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Theorem cofsmo 9129
 Description: Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypotheses
Ref Expression
cofsmo.1 𝐶 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}
cofsmo.2 𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
cofsmo.3 𝑂 = OrdIso( E , 𝐶)
Assertion
Ref Expression
cofsmo ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
Distinct variable groups:   𝑓,𝑔,𝑣,𝑤,𝑥,𝑧,𝐴   𝑦,𝑓,𝐵,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶   𝑣,𝐾,𝑤,𝑦   𝑔,𝑂,𝑣,𝑥,𝑧
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑔)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)   𝐾(𝑥,𝑧,𝑓,𝑔)   𝑂(𝑦,𝑤,𝑓)

Proof of Theorem cofsmo
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofsmo.1 . . . . . . . . . . . . 13 𝐶 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}
2 ssrab2 3720 . . . . . . . . . . . . 13 {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)} ⊆ 𝐵
31, 2eqsstri 3668 . . . . . . . . . . . 12 𝐶𝐵
4 ssexg 4837 . . . . . . . . . . . 12 ((𝐶𝐵𝐵 ∈ On) → 𝐶 ∈ V)
53, 4mpan 706 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐶 ∈ V)
6 onss 7032 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ⊆ On)
73, 6syl5ss 3647 . . . . . . . . . . . 12 (𝐵 ∈ On → 𝐶 ⊆ On)
8 epweon 7025 . . . . . . . . . . . 12 E We On
9 wess 5130 . . . . . . . . . . . 12 (𝐶 ⊆ On → ( E We On → E We 𝐶))
107, 8, 9mpisyl 21 . . . . . . . . . . 11 (𝐵 ∈ On → E We 𝐶)
11 cofsmo.3 . . . . . . . . . . . 12 𝑂 = OrdIso( E , 𝐶)
1211oiiso 8483 . . . . . . . . . . 11 ((𝐶 ∈ V ∧ E We 𝐶) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
135, 10, 12syl2anc 694 . . . . . . . . . 10 (𝐵 ∈ On → 𝑂 Isom E , E (dom 𝑂, 𝐶))
1413ad2antlr 763 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
15 isof1o 6613 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, 𝐶) → 𝑂:dom 𝑂1-1-onto𝐶)
16 f1ofo 6182 . . . . . . . . 9 (𝑂:dom 𝑂1-1-onto𝐶𝑂:dom 𝑂onto𝐶)
1714, 15, 163syl 18 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂onto𝐶)
18 fof 6153 . . . . . . . . 9 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐶)
19 fss 6094 . . . . . . . . 9 ((𝑂:dom 𝑂𝐶𝐶𝐵) → 𝑂:dom 𝑂𝐵)
2018, 3, 19sylancl 695 . . . . . . . 8 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐵)
2117, 20syl 17 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐵)
2211oion 8482 . . . . . . . . . 10 (𝐶 ∈ V → dom 𝑂 ∈ On)
235, 22syl 17 . . . . . . . . 9 (𝐵 ∈ On → dom 𝑂 ∈ On)
2423ad2antlr 763 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ On)
25 simplr 807 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐵 ∈ On)
26 eloni 5771 . . . . . . . . . . 11 (dom 𝑂 ∈ On → Ord dom 𝑂)
27 smoiso2 7511 . . . . . . . . . . 11 ((Ord dom 𝑂𝐶 ⊆ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2826, 7, 27syl2an 493 . . . . . . . . . 10 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2928biimpar 501 . . . . . . . . 9 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → (𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂))
3029simprd 478 . . . . . . . 8 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → Smo 𝑂)
3124, 25, 14, 30syl21anc 1365 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo 𝑂)
32 eloni 5771 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
3332ad2antlr 763 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord 𝐵)
34 smorndom 7510 . . . . . . 7 ((𝑂:dom 𝑂𝐵 ∧ Smo 𝑂 ∧ Ord 𝐵) → dom 𝑂𝐵)
3521, 31, 33, 34syl3anc 1366 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂𝐵)
36 onsssuc 5851 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3724, 25, 36syl2anc 694 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3835, 37mpbid 222 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ suc 𝐵)
3938adantrr 753 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → dom 𝑂 ∈ suc 𝐵)
40 vex 3234 . . . . . 6 𝑓 ∈ V
4111oiexg 8481 . . . . . . . 8 (𝐶 ∈ V → 𝑂 ∈ V)
425, 41syl 17 . . . . . . 7 (𝐵 ∈ On → 𝑂 ∈ V)
4342ad2antlr 763 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂 ∈ V)
44 coexg 7159 . . . . . 6 ((𝑓 ∈ V ∧ 𝑂 ∈ V) → (𝑓𝑂) ∈ V)
4540, 43, 44sylancr 696 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂) ∈ V)
46 simprl 809 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑓:𝐵𝐴)
4721adantrr 753 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂:dom 𝑂𝐵)
48 fco 6096 . . . . . . 7 ((𝑓:𝐵𝐴𝑂:dom 𝑂𝐵) → (𝑓𝑂):dom 𝑂𝐴)
4946, 47, 48syl2anc 694 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂):dom 𝑂𝐴)
50 simpr 476 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑓:𝐵𝐴)
5150, 21, 48syl2anc 694 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑓𝑂):dom 𝑂𝐴)
52 ordsson 7031 . . . . . . . . 9 (Ord 𝐴𝐴 ⊆ On)
5352ad2antrr 762 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐴 ⊆ On)
5424, 26syl 17 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord dom 𝑂)
5517, 18syl 17 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐶)
56 simpl 472 . . . . . . . . . . . 12 ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑠 ∈ dom 𝑂)
57 ffvelrn 6397 . . . . . . . . . . . 12 ((𝑂:dom 𝑂𝐶𝑠 ∈ dom 𝑂) → (𝑂𝑠) ∈ 𝐶)
5855, 56, 57syl2an 493 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑠) ∈ 𝐶)
59 ffn 6083 . . . . . . . . . . . . . 14 (𝑂:dom 𝑂𝐶𝑂 Fn dom 𝑂)
6017, 18, 593syl 18 . . . . . . . . . . . . 13 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Fn dom 𝑂)
6160, 31jca 553 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑂 Fn dom 𝑂 ∧ Smo 𝑂))
62 smoel2 7505 . . . . . . . . . . . 12 (((𝑂 Fn dom 𝑂 ∧ Smo 𝑂) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
6361, 62sylan 487 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
64 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑂𝑠) → (𝑓𝑧) = (𝑓‘(𝑂𝑠)))
6564eleq2d 2716 . . . . . . . . . . . . . . 15 (𝑧 = (𝑂𝑠) → ((𝑓𝑥) ∈ (𝑓𝑧) ↔ (𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
6665raleqbi1dv 3176 . . . . . . . . . . . . . 14 (𝑧 = (𝑂𝑠) → (∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧) ↔ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
67 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → (𝑓𝑤) = (𝑓𝑥))
6867eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑦)))
6968cbvralv 3201 . . . . . . . . . . . . . . . . 17 (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦))
70 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
7170eleq2d 2716 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝑓𝑥) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑧)))
7271raleqbi1dv 3176 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7369, 72syl5bb 272 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7473cbvrabv 3230 . . . . . . . . . . . . . . 15 {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)} = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
751, 74eqtri 2673 . . . . . . . . . . . . . 14 𝐶 = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
7666, 75elrab2 3399 . . . . . . . . . . . . 13 ((𝑂𝑠) ∈ 𝐶 ↔ ((𝑂𝑠) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
7776simprbi 479 . . . . . . . . . . . 12 ((𝑂𝑠) ∈ 𝐶 → ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)))
78 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = (𝑂𝑡) → (𝑓𝑥) = (𝑓‘(𝑂𝑡)))
7978eleq1d 2715 . . . . . . . . . . . . 13 (𝑥 = (𝑂𝑡) → ((𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) ↔ (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8079rspccv 3337 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8177, 80syl 17 . . . . . . . . . . 11 ((𝑂𝑠) ∈ 𝐶 → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8258, 63, 81sylc 65 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠)))
8321adantr 480 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑂:dom 𝑂𝐵)
84 ordtr1 5805 . . . . . . . . . . . . . 14 (Ord dom 𝑂 → ((𝑡𝑠𝑠 ∈ dom 𝑂) → 𝑡 ∈ dom 𝑂))
8584ancomsd 469 . . . . . . . . . . . . 13 (Ord dom 𝑂 → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8624, 26, 853syl 18 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8786imp 444 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑡 ∈ dom 𝑂)
88 fvco3 6314 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑡 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
8983, 87, 88syl2anc 694 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
90 simprl 809 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑠 ∈ dom 𝑂)
91 fvco3 6314 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑠 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
9283, 90, 91syl2anc 694 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
9382, 89, 923eltr4d 2745 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
9493ralrimivva 3000 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
95 issmo2 7491 . . . . . . . . 9 ((𝑓𝑂):dom 𝑂𝐴 → ((𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠)) → Smo (𝑓𝑂)))
9695imp 444 . . . . . . . 8 (((𝑓𝑂):dom 𝑂𝐴 ∧ (𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))) → Smo (𝑓𝑂))
9751, 53, 54, 94, 96syl13anc 1368 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo (𝑓𝑂))
9897adantrr 753 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → Smo (𝑓𝑂))
9917adantr 480 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝑂:dom 𝑂onto𝐶)
100 rabn0 3991 . . . . . . . . . . . . . . . . . 18 ({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅ ↔ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤))
101 ssrab2 3720 . . . . . . . . . . . . . . . . . . . 20 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ 𝐵
102101, 6syl5ss 3647 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ On → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
103 cofsmo.2 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
104 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
105104sseq2d 3666 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓𝑤)))
106105cbvrabv 3230 . . . . . . . . . . . . . . . . . . . . . 22 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
107106inteqi 4511 . . . . . . . . . . . . . . . . . . . . 21 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
108103, 107eqtri 2673 . . . . . . . . . . . . . . . . . . . 20 𝐾 = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
109 onint 7037 . . . . . . . . . . . . . . . . . . . 20 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
110108, 109syl5eqel 2734 . . . . . . . . . . . . . . . . . . 19 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
111102, 110sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
112100, 111sylan2br 492 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
113 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝐾 → (𝑓𝑤) = (𝑓𝐾))
114113sseq2d 3666 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝐾 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝐾)))
115114elrab 3396 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
116112, 115sylib 208 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
117116ex 449 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
118117adantl 481 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
119 simpr2 1088 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐵)
120 simp3 1083 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐾)
121108eleq2i 2722 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝐾𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
122 simp21 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵𝐴)
123 simp1l 1105 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → Ord 𝐴)
124123, 52syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐴 ⊆ On)
125122, 124fssd 6095 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵⟶On)
126 simp22 1115 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐾𝐵)
127125, 126ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝐾) ∈ On)
128 simp1r 1106 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐵 ∈ On)
129 ontr1 5809 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → ((𝑤𝐾𝐾𝐵) → 𝑤𝐵))
1301293impib 1281 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) → 𝑤𝐵)
131128, 120, 126, 130syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐵)
132125, 131ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ On)
133 ontri1 5795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓𝐾) ∈ On ∧ (𝑓𝑤) ∈ On) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
134127, 132, 133syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
135 simp23 1116 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑧 ⊆ (𝑓𝐾))
136 simpl1 1084 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝐵 ∈ On)
137136, 102syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
138 sstr 3644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤)) → 𝑧 ⊆ (𝑓𝑤))
139130, 138anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
140 rabid 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
141139, 140sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
142 onnmin 7045 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
143137, 141, 142syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
144143expr 642 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ 𝑧 ⊆ (𝑓𝐾)) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
145128, 120, 126, 135, 144syl31anc 1369 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
146134, 145sylbird 250 . . . . . . . . . . . . . . . . . . . . . . 23 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (¬ (𝑓𝑤) ∈ (𝑓𝐾) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
147146con4d 114 . . . . . . . . . . . . . . . . . . . . . 22 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝑓𝑤) ∈ (𝑓𝐾)))
148121, 147syl5bi 232 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
149120, 148mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ (𝑓𝐾))
1501493expia 1286 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
151150ralrimiv 2994 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾))
152 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐾 → (𝑓𝑦) = (𝑓𝐾))
153152eleq2d 2716 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐾 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑤) ∈ (𝑓𝐾)))
154153raleqbi1dv 3176 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐾 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
155154, 1elrab2 3399 . . . . . . . . . . . . . . . . . 18 (𝐾𝐶 ↔ (𝐾𝐵 ∧ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
156119, 151, 155sylanbrc 699 . . . . . . . . . . . . . . . . 17 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐶)
157156expcom 450 . . . . . . . . . . . . . . . 16 ((𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶))
1581573expib 1287 . . . . . . . . . . . . . . 15 (𝑓:𝐵𝐴 → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶)))
159158com13 88 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → (𝑓:𝐵𝐴𝐾𝐶)))
160118, 159syld 47 . . . . . . . . . . . . 13 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝑓:𝐵𝐴𝐾𝐶)))
161160com23 86 . . . . . . . . . . . 12 ((Ord 𝐴𝐵 ∈ On) → (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → 𝐾𝐶)))
162161imp31 447 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾𝐶)
163 foelrn 6418 . . . . . . . . . . 11 ((𝑂:dom 𝑂onto𝐶𝐾𝐶) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
16499, 162, 163syl2anc 694 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
165 simpllr 815 . . . . . . . . . . . . . 14 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐵 ∈ On)
166 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝐾 = (𝑂𝑣) → (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
167166biimpcd 239 . . . . . . . . . . . . . . . 16 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
168 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑂𝑣) → (𝑓𝑥) = (𝑓‘(𝑂𝑣)))
169168sseq2d 3666 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑂𝑣) → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
17067sseq2d 3666 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝑥)))
171170cbvrabv 3230 . . . . . . . . . . . . . . . . . 18 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
172169, 171elrab2 3399 . . . . . . . . . . . . . . . . 17 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ ((𝑂𝑣) ∈ 𝐵𝑧 ⊆ (𝑓‘(𝑂𝑣))))
173172simprbi 479 . . . . . . . . . . . . . . . 16 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → 𝑧 ⊆ (𝑓‘(𝑂𝑣)))
174167, 173syl6 35 . . . . . . . . . . . . . . 15 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
175112, 174syl 17 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
176165, 175sylancom 702 . . . . . . . . . . . . 13 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
177176adantr 480 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
17821ad2antrr 762 . . . . . . . . . . . . . 14 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → 𝑂:dom 𝑂𝐵)
179 fvco3 6314 . . . . . . . . . . . . . 14 ((𝑂:dom 𝑂𝐵𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
180178, 179sylancom 702 . . . . . . . . . . . . 13 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
181180sseq2d 3666 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝑧 ⊆ ((𝑓𝑂)‘𝑣) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
182177, 181sylibrd 249 . . . . . . . . . . 11 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
183182reximdva 3046 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
184164, 183mpd 15 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
185184ex 449 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
186185ralimdv 2992 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
187186impr 648 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
18849, 98, 1873jca 1261 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
189 feq1 6064 . . . . . . 7 (𝑔 = (𝑓𝑂) → (𝑔:dom 𝑂𝐴 ↔ (𝑓𝑂):dom 𝑂𝐴))
190 smoeq 7492 . . . . . . 7 (𝑔 = (𝑓𝑂) → (Smo 𝑔 ↔ Smo (𝑓𝑂)))
191 fveq1 6228 . . . . . . . . . 10 (𝑔 = (𝑓𝑂) → (𝑔𝑣) = ((𝑓𝑂)‘𝑣))
192191sseq2d 3666 . . . . . . . . 9 (𝑔 = (𝑓𝑂) → (𝑧 ⊆ (𝑔𝑣) ↔ 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
193192rexbidv 3081 . . . . . . . 8 (𝑔 = (𝑓𝑂) → (∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
194193ralbidv 3015 . . . . . . 7 (𝑔 = (𝑓𝑂) → (∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
195189, 190, 1943anbi123d 1439 . . . . . 6 (𝑔 = (𝑓𝑂) → ((𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)) ↔ ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))))
196195spcegv 3325 . . . . 5 ((𝑓𝑂) ∈ V → (((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)) → ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
19745, 188, 196sylc 65 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
198 feq2 6065 . . . . . . 7 (𝑥 = dom 𝑂 → (𝑔:𝑥𝐴𝑔:dom 𝑂𝐴))
199 rexeq 3169 . . . . . . . 8 (𝑥 = dom 𝑂 → (∃𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
200199ralbidv 3015 . . . . . . 7 (𝑥 = dom 𝑂 → (∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
201198, 2003anbi13d 1441 . . . . . 6 (𝑥 = dom 𝑂 → ((𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ (𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
202201exbidv 1890 . . . . 5 (𝑥 = dom 𝑂 → (∃𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
203202rspcev 3340 . . . 4 ((dom 𝑂 ∈ suc 𝐵 ∧ ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)))
20439, 197, 203syl2anc 694 . . 3 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)))
205204ex 449 . 2 ((Ord 𝐴𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
206205exlimdv 1901 1 ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ∩ cint 4507   E cep 5057   We wwe 5101  dom cdm 5143   ∘ ccom 5147  Ord word 5760  Oncon0 5761  suc csuc 5763   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924  –1-1-onto→wf1o 5925  ‘cfv 5926   Isom wiso 5927  Smo wsmo 7487  OrdIsocoi 8455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-smo 7488  df-recs 7513  df-oi 8456 This theorem is referenced by:  cfcof  9134
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