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Theorem fin23lem34 9206
Description: Lemma for fin23 9249. Establish induction invariants on 𝑌 which parameterizes our contradictory chain of subsets. In this section, is the hypothetically assumed family of subsets, 𝑔 is the ground set, and 𝑖 is the induction function constructed in the previous section. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem33.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.f (𝜑:ω–1-1→V)
fin23lem.g (𝜑 ran 𝐺)
fin23lem.h (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
fin23lem.i 𝑌 = (rec(𝑖, ) ↾ ω)
Assertion
Ref Expression
fin23lem34 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐴(𝑥,𝑔,,𝑖)   𝐹(𝑥,𝑔,,𝑖,𝑗)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem34
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . . 6 (𝑎 = ∅ → (𝑌𝑎) = (𝑌‘∅))
2 f1eq1 6134 . . . . . 6 ((𝑌𝑎) = (𝑌‘∅) → ((𝑌𝑎):ω–1-1→V ↔ (𝑌‘∅):ω–1-1→V))
31, 2syl 17 . . . . 5 (𝑎 = ∅ → ((𝑌𝑎):ω–1-1→V ↔ (𝑌‘∅):ω–1-1→V))
41rneqd 5385 . . . . . . 7 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
54unieqd 4478 . . . . . 6 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
65sseq1d 3665 . . . . 5 (𝑎 = ∅ → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘∅) ⊆ 𝐺))
73, 6anbi12d 747 . . . 4 (𝑎 = ∅ → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺)))
87imbi2d 329 . . 3 (𝑎 = ∅ → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺))))
9 fveq2 6229 . . . . . 6 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
10 f1eq1 6134 . . . . . 6 ((𝑌𝑎) = (𝑌𝑏) → ((𝑌𝑎):ω–1-1→V ↔ (𝑌𝑏):ω–1-1→V))
119, 10syl 17 . . . . 5 (𝑎 = 𝑏 → ((𝑌𝑎):ω–1-1→V ↔ (𝑌𝑏):ω–1-1→V))
129rneqd 5385 . . . . . . 7 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
1312unieqd 4478 . . . . . 6 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
1413sseq1d 3665 . . . . 5 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝑏) ⊆ 𝐺))
1511, 14anbi12d 747 . . . 4 (𝑎 = 𝑏 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
1615imbi2d 329 . . 3 (𝑎 = 𝑏 → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺))))
17 fveq2 6229 . . . . . 6 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
18 f1eq1 6134 . . . . . 6 ((𝑌𝑎) = (𝑌‘suc 𝑏) → ((𝑌𝑎):ω–1-1→V ↔ (𝑌‘suc 𝑏):ω–1-1→V))
1917, 18syl 17 . . . . 5 (𝑎 = suc 𝑏 → ((𝑌𝑎):ω–1-1→V ↔ (𝑌‘suc 𝑏):ω–1-1→V))
2017rneqd 5385 . . . . . . 7 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2120unieqd 4478 . . . . . 6 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2221sseq1d 3665 . . . . 5 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘suc 𝑏) ⊆ 𝐺))
2319, 22anbi12d 747 . . . 4 (𝑎 = suc 𝑏 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺)))
2423imbi2d 329 . . 3 (𝑎 = suc 𝑏 → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
25 fveq2 6229 . . . . . 6 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
26 f1eq1 6134 . . . . . 6 ((𝑌𝑎) = (𝑌𝐴) → ((𝑌𝑎):ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
2725, 26syl 17 . . . . 5 (𝑎 = 𝐴 → ((𝑌𝑎):ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
2825rneqd 5385 . . . . . . 7 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
2928unieqd 4478 . . . . . 6 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
3029sseq1d 3665 . . . . 5 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝐴) ⊆ 𝐺))
3127, 30anbi12d 747 . . . 4 (𝑎 = 𝐴 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
3231imbi2d 329 . . 3 (𝑎 = 𝐴 → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))))
33 fin23lem.f . . . 4 (𝜑:ω–1-1→V)
34 fin23lem.g . . . 4 (𝜑 ran 𝐺)
35 fin23lem.i . . . . . . . 8 𝑌 = (rec(𝑖, ) ↾ ω)
3635fveq1i 6230 . . . . . . 7 (𝑌‘∅) = ((rec(𝑖, ) ↾ ω)‘∅)
37 vex 3234 . . . . . . . 8 ∈ V
38 fr0g 7576 . . . . . . . 8 ( ∈ V → ((rec(𝑖, ) ↾ ω)‘∅) = )
3937, 38ax-mp 5 . . . . . . 7 ((rec(𝑖, ) ↾ ω)‘∅) =
4036, 39eqtri 2673 . . . . . 6 (𝑌‘∅) =
41 f1eq1 6134 . . . . . 6 ((𝑌‘∅) = → ((𝑌‘∅):ω–1-1→V ↔ :ω–1-1→V))
4240, 41ax-mp 5 . . . . 5 ((𝑌‘∅):ω–1-1→V ↔ :ω–1-1→V)
4340rneqi 5384 . . . . . . 7 ran (𝑌‘∅) = ran
4443unieqi 4477 . . . . . 6 ran (𝑌‘∅) = ran
4544sseq1i 3662 . . . . 5 ( ran (𝑌‘∅) ⊆ 𝐺 ran 𝐺)
4642, 45anbi12i 733 . . . 4 (((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺) ↔ (:ω–1-1→V ∧ ran 𝐺))
4733, 34, 46sylanbrc 699 . . 3 (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺))
48 fin23lem.h . . . . . . . . . 10 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
49 fvex 6239 . . . . . . . . . . 11 (𝑌𝑏) ∈ V
50 f1eq1 6134 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → (𝑗:ω–1-1→V ↔ (𝑌𝑏):ω–1-1→V))
51 rneq 5383 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5251unieqd 4478 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5352sseq1d 3665 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran 𝑗𝐺 ran (𝑌𝑏) ⊆ 𝐺))
5450, 53anbi12d 747 . . . . . . . . . . . 12 (𝑗 = (𝑌𝑏) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
55 fveq2 6229 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → (𝑖𝑗) = (𝑖‘(𝑌𝑏)))
56 f1eq1 6134 . . . . . . . . . . . . . 14 ((𝑖𝑗) = (𝑖‘(𝑌𝑏)) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
5755, 56syl 17 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
5855rneqd 5385 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
5958unieqd 4478 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
6059, 52psseq12d 3734 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
6157, 60anbi12d 747 . . . . . . . . . . . 12 (𝑗 = (𝑌𝑏) → (((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗) ↔ ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏))))
6254, 61imbi12d 333 . . . . . . . . . . 11 (𝑗 = (𝑌𝑏) → (((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)) ↔ (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))))
6349, 62spcv 3330 . . . . . . . . . 10 (∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)) → (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏))))
6448, 63syl 17 . . . . . . . . 9 (𝜑 → (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏))))
6564imp 444 . . . . . . . 8 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
66 pssss 3735 . . . . . . . . . . . 12 ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏))
67 sstr 3644 . . . . . . . . . . . 12 (( ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6866, 67sylan 487 . . . . . . . . . . 11 (( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6968expcom 450 . . . . . . . . . 10 ( ran (𝑌𝑏) ⊆ 𝐺 → ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
7069anim2d 588 . . . . . . . . 9 ( ran (𝑌𝑏) ⊆ 𝐺 → (((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)))
7170ad2antll 765 . . . . . . . 8 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → (((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)))
7265, 71mpd 15 . . . . . . 7 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
73723adant1 1099 . . . . . 6 ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
74 frsuc 7577 . . . . . . . . 9 (𝑏 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝑏) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏)))
7535fveq1i 6230 . . . . . . . . 9 (𝑌‘suc 𝑏) = ((rec(𝑖, ) ↾ ω)‘suc 𝑏)
7635fveq1i 6230 . . . . . . . . . 10 (𝑌𝑏) = ((rec(𝑖, ) ↾ ω)‘𝑏)
7776fveq2i 6232 . . . . . . . . 9 (𝑖‘(𝑌𝑏)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏))
7874, 75, 773eqtr4g 2710 . . . . . . . 8 (𝑏 ∈ ω → (𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)))
79 f1eq1 6134 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ((𝑌‘suc 𝑏):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
80 rneq 5383 . . . . . . . . . . 11 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8180unieqd 4478 . . . . . . . . . 10 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8281sseq1d 3665 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ( ran (𝑌‘suc 𝑏) ⊆ 𝐺 ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
8379, 82anbi12d 747 . . . . . . . 8 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)))
8478, 83syl 17 . . . . . . 7 (𝑏 ∈ ω → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)))
85843ad2ant1 1102 . . . . . 6 ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → (((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺) ↔ ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)))
8673, 85mpbird 247 . . . . 5 ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))
87863exp 1283 . . . 4 (𝑏 ∈ ω → (𝜑 → (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
8887a2d 29 . . 3 (𝑏 ∈ ω → ((𝜑 → ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → (𝜑 → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
898, 16, 24, 32, 47, 88finds 7134 . 2 (𝐴 ∈ ω → (𝜑 → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
9089impcom 445 1 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  wss 3607  wpss 3608  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507  ran crn 5144  cres 5145  suc csuc 5763  1-1wf1 5923  cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550  𝑚 cmap 7899
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-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-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-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-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-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  fin23lem35  9207  fin23lem39  9210
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