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Theorem dffi3 9426
Description: The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
dffi3.1 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
Assertion
Ref Expression
dffi3 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉   𝑦,𝑢,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑢)   𝑅(𝑧,𝑢)   𝑉(𝑧,𝑢)

Proof of Theorem dffi3
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffi2 9418 . . . 4 (𝐴𝑉 → (fi‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
2 fr0g 8436 . . . . . . . 8 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3 frfnom 8435 . . . . . . . . 9 (rec(𝑅, 𝐴) ↾ ω) Fn ω
4 peano1 7879 . . . . . . . . 9 ∅ ∈ ω
5 fnfvelrn 7083 . . . . . . . . 9 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
63, 4, 5mp2an 691 . . . . . . . 8 ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
72, 6eqeltrrdi 2843 . . . . . . 7 (𝐴𝑉𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8 elssuni 4942 . . . . . . 7 (𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω) → 𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
97, 8syl 17 . . . . . 6 (𝐴𝑉𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
10 reeanv 3227 . . . . . . . . 9 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
11 eliun 5002 . . . . . . . . . 10 (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12 eliun 5002 . . . . . . . . . 10 (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
1311, 12anbi12i 628 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
14 fniunfv 7246 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ran (rec(𝑅, 𝐴) ↾ ω))
1514eleq2d 2820 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ran (rec(𝑅, 𝐴) ↾ ω)))
16 fniunfv 7246 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ran (rec(𝑅, 𝐴) ↾ ω))
1716eleq2d 2820 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
1815, 17anbi12d 632 . . . . . . . . . 10 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))))
193, 18ax-mp 5 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
2010, 13, 193bitr2i 299 . . . . . . . 8 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
21 ordom 7865 . . . . . . . . . . . . . . . 16 Ord ω
22 ordunel 7815 . . . . . . . . . . . . . . . 16 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2321, 22mp3an1 1449 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2423adantl 483 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚𝑛) ∈ ω)
25 simprl 770 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
2624, 25jca 513 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27 nnon 7861 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ω → 𝑦 ∈ On)
28 nnon 7861 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ω → 𝑥 ∈ On)
2928ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
30 onsseleq 6406 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
3127, 29, 30syl2an2 685 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
32 rzal 4509 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
3332biantrud 533 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
34 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
3534sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
3633, 35bitr3d 281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ∅ → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
37 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
3837sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
3937sseq2d 4015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4039raleqbi1dv 3334 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4138, 40anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))))
42 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
4342sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
4442sseq2d 4015 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4544raleqbi1dv 3334 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4643, 45anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = suc 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
47 ssfii 9414 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
482, 47eqsstrd 4021 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴))
49 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
50 eqidd 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
51 ineq1 4206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = 𝑥 → (𝑎𝑏) = (𝑥𝑏))
5251eqeq2d 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑥 → (𝑥 = (𝑎𝑏) ↔ 𝑥 = (𝑥𝑏)))
53 ineq2 4207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑥 → (𝑥𝑏) = (𝑥𝑥))
54 inidm 4219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑥) = 𝑥
5553, 54eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = 𝑥 → (𝑥𝑏) = 𝑥)
5655eqeq2d 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑥 → (𝑥 = (𝑥𝑏) ↔ 𝑥 = 𝑥))
5752, 56rspc2ev 3625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 = 𝑥) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
5849, 49, 50, 57syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
59 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
6059rnmpo 7542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏)}
6160eqabri 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
6258, 61sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
6362ssriv 3987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
64 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
65 fvex 6905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6665uniex 7731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6766pwex 5379 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
68 inss1 4229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ⊆ 𝑎
69 elssuni 4942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7069adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7168, 70sstrid 3994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
72 vex 3479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑎 ∈ V
7372inex1 5318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ∈ V
7473elpw 4607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7571, 74sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7675rgen2 3198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
7759fmpo 8054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7876, 77mpbi 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
79 frn 6725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
8078, 79ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
8167, 80ssexi 5323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V
82 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝐴
83 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝑛
84 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
85 dffi3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
86 mpoeq12 7482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑢 = 𝑣𝑢 = 𝑣) → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
8786anidms 568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
88 ineq1 4206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 = 𝑎 → (𝑦𝑧) = (𝑎𝑧))
89 ineq2 4207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧 = 𝑏 → (𝑎𝑧) = (𝑎𝑏))
9088, 89cbvmpov 7504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))
9187, 90eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9291rneqd 5938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑢 = 𝑣 → ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9392cbvmptv 5262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧))) = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9485, 93eqtri 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
95 rdgeq1 8411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))) → rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴))
9694, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴)
9796reseq1i 5978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴) ↾ ω)
98 mpoeq12 7482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
9998anidms 568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10099rneqd 5938 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10182, 83, 84, 97, 100frsucmpt 8438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10264, 81, 101sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10363, 102sseqtrrid 4036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
104 sstr2 3990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
105103, 104syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
106105ralimdv 3170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
107 vex 3479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑛 ∈ V
108 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
109108sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
110107, 109ralsn 4686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
111103, 110sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
112106, 111jctird 528 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
113 df-suc 6371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 suc 𝑛 = (𝑛 ∪ {𝑛})
114113raleqi 3324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
115 ralunb 4192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
116114, 115bitri 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
117112, 116imbitrrdi 251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
118 fiin 9417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎𝑏) ∈ (fi‘𝐴))
119118rgen2 3198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴)
120 ss2ralv 4053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
121119, 120mpi 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴))
12259fmpo 8054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
123121, 122sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
124123frnd 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
125124adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
126102, 125eqsstrd 4021 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
127117, 126jctild 527 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
128127expimpd 455 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
129128a1d 25 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → (𝐴𝑉 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))))
13036, 41, 46, 48, 129finds2 7891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ω → (𝐴𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
131130impcom 409 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
132131simprd 497 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
133132r19.21bi 3249 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
134133ex 414 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝑥 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
135134adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
136 fveq2 6892 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
137 eqimss 4041 . . . . . . . . . . . . . . . . . . . 20 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
138136, 137syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
139138a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
140135, 139jaod 858 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦𝑥𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
14131, 140sylbid 239 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
142141ralrimiva 3147 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
143142ralrimiva 3147 . . . . . . . . . . . . . 14 (𝐴𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
144143adantr 482 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
145 ssun1 4173 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
146145a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚𝑛))
147 sseq2 4009 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (𝑦𝑥𝑦 ⊆ (𝑚𝑛)))
148 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
149148sseq2d 4015 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
150147, 149imbi12d 345 . . . . . . . . . . . . . 14 (𝑥 = (𝑚𝑛) → ((𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
151 sseq1 4008 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑚 ⊆ (𝑚𝑛)))
152 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
153152sseq1d 4014 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
154151, 153imbi12d 345 . . . . . . . . . . . . . 14 (𝑦 = 𝑚 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
155150, 154rspc2v 3623 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
15626, 144, 146, 155syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
157156sseld 3982 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
158 simprr 772 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
15924, 158jca 513 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω))
160 ssun2 4174 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
161160a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚𝑛))
162 sseq1 4008 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑛 ⊆ (𝑚𝑛)))
163108sseq1d 4014 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
164162, 163imbi12d 345 . . . . . . . . . . . . . 14 (𝑦 = 𝑛 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
165150, 164rspc2v 3623 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
166159, 144, 161, 165syl3c 66 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
167166sseld 3982 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
16823ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑚𝑛) ∈ ω)
169 peano2 7881 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ ω → suc (𝑚𝑛) ∈ ω)
170 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑥 = suc (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
171170ssiun2s 5052 . . . . . . . . . . . . . . 15 (suc (𝑚𝑛) ∈ ω → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
172168, 169, 1713syl 18 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
173 simprl 770 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
174 simprr 772 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
175 eqidd 2734 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) = (𝑐𝑑))
176 ineq1 4206 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
177176eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ((𝑐𝑑) = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑐𝑏)))
178 ineq2 4207 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
179178eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → ((𝑐𝑑) = (𝑐𝑏) ↔ (𝑐𝑑) = (𝑐𝑑)))
180177, 179rspc2ev 3625 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ (𝑐𝑑) = (𝑐𝑑)) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
181173, 174, 175, 180syl3anc 1372 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
182 vex 3479 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
183182inex1 5318 . . . . . . . . . . . . . . . . . 18 (𝑐𝑑) ∈ V
184 eqeq1 2737 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑐𝑑) → (𝑥 = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑎𝑏)))
1851842rexbidv 3220 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑐𝑑) → (∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏)))
186183, 185elab 3669 . . . . . . . . . . . . . . . . 17 ((𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)} ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
187181, 186sylibr 233 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)})
188 eqid 2733 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
189188rnmpo 7542 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)}
190187, 189eleqtrrdi 2845 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
191 fvex 6905 . . . . . . . . . . . . . . . . . . 19 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
192191uniex 7731 . . . . . . . . . . . . . . . . . 18 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
193192pwex 5379 . . . . . . . . . . . . . . . . 17 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
194 elssuni 4942 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19568, 194sstrid 3994 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
19673elpw 4607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
197195, 196sylibr 233 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
198197adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
199198rgen2 3198 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
200188fmpo 8054 . . . . . . . . . . . . . . . . . . 19 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
201199, 200mpbi 229 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
202 frn 6725 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
203201, 202ax-mp 5 . . . . . . . . . . . . . . . . 17 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
204193, 203ssexi 5323 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V
205 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑣(𝑚𝑛)
206 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
207 mpoeq12 7482 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
208207anidms 568 . . . . . . . . . . . . . . . . . 18 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
209208rneqd 5938 . . . . . . . . . . . . . . . . 17 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
21082, 205, 206, 97, 209frsucmpt 8438 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
211168, 204, 210sylancl 587 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
212190, 211eleqtrrd 2837 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
213172, 212sseldd 3984 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
214 fniunfv 7246 . . . . . . . . . . . . . 14 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω))
2153, 214ax-mp 5 . . . . . . . . . . . . 13 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω)
216213, 215eleqtrdi 2844 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
217216ex 414 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
218157, 167, 217syl2and 609 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
219218rexlimdvva 3212 . . . . . . . . 9 (𝐴𝑉 → (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
220219imp 408 . . . . . . . 8 ((𝐴𝑉 ∧ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
22120, 220sylan2br 596 . . . . . . 7 ((𝐴𝑉 ∧ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
222221ralrimivva 3201 . . . . . 6 (𝐴𝑉 → ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
223131simpld 496 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
224 fvex 6905 . . . . . . . . . . . . 13 (fi‘𝐴) ∈ V
225224elpw2 5346 . . . . . . . . . . . 12 (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
226223, 225sylibr 233 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
227226ralrimiva 3147 . . . . . . . . . 10 (𝐴𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
228 fnfvrnss 7120 . . . . . . . . . 10 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴)) → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
2293, 227, 228sylancr 588 . . . . . . . . 9 (𝐴𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
230 sspwuni 5104 . . . . . . . . 9 (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
231229, 230sylib 217 . . . . . . . 8 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
232 ssexg 5324 . . . . . . . 8 (( ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
233231, 224, 232sylancl 587 . . . . . . 7 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
234 sseq2 4009 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴𝑥𝐴 ran (rec(𝑅, 𝐴) ↾ ω)))
235 eleq2 2823 . . . . . . . . . . 11 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐𝑑) ∈ 𝑥 ↔ (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
236235raleqbi1dv 3334 . . . . . . . . . 10 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
237236raleqbi1dv 3334 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
238234, 237anbi12d 632 . . . . . . . 8 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥) ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
239238elabg 3667 . . . . . . 7 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ V → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
240233, 239syl 17 . . . . . 6 (𝐴𝑉 → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
2419, 222, 240mpbir2and 712 . . . . 5 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
242 intss1 4968 . . . . 5 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
243241, 242syl 17 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
2441, 243eqsstrd 4021 . . 3 (𝐴𝑉 → (fi‘𝐴) ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
245244, 231eqssd 4000 . 2 (𝐴𝑉 → (fi‘𝐴) = ran (rec(𝑅, 𝐴) ↾ ω))
246 df-ima 5690 . . 3 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
247246unieqi 4922 . 2 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
248245, 247eqtr4di 2791 1 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  cun 3947  cin 3948  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629   cuni 4909   cint 4951   ciun 4998  cmpt 5232   × cxp 5675  ran crn 5678  cres 5679  cima 5680  Ord word 6364  Oncon0 6365  suc csuc 6367   Fn wfn 6539  wf 6540  cfv 6544  cmpo 7411  ωcom 7855  reccrdg 8409  ficfi 9405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406
This theorem is referenced by: (None)
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