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Theorem fin23lem34 9839
Description: Lemma for fin23 9882. 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 6668 . . . . . 6 (𝑎 = ∅ → (𝑌𝑎) = (𝑌‘∅))
2 f1eq1 6563 . . . . . 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 5775 . . . . . . 7 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
54unieqd 4807 . . . . . 6 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
65sseq1d 3906 . . . . 5 (𝑎 = ∅ → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘∅) ⊆ 𝐺))
73, 6anbi12d 634 . . . 4 (𝑎 = ∅ → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺)))
87imbi2d 344 . . 3 (𝑎 = ∅ → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺))))
9 fveq2 6668 . . . . . 6 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
10 f1eq1 6563 . . . . . 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 5775 . . . . . . 7 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
1312unieqd 4807 . . . . . 6 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
1413sseq1d 3906 . . . . 5 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝑏) ⊆ 𝐺))
1511, 14anbi12d 634 . . . 4 (𝑎 = 𝑏 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
1615imbi2d 344 . . 3 (𝑎 = 𝑏 → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺))))
17 fveq2 6668 . . . . . 6 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
18 f1eq1 6563 . . . . . 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 5775 . . . . . . 7 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2120unieqd 4807 . . . . . 6 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2221sseq1d 3906 . . . . 5 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘suc 𝑏) ⊆ 𝐺))
2319, 22anbi12d 634 . . . 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 344 . . 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 6668 . . . . . 6 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
26 f1eq1 6563 . . . . . 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 5775 . . . . . . 7 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
2928unieqd 4807 . . . . . 6 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
3029sseq1d 3906 . . . . 5 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝐴) ⊆ 𝐺))
3127, 30anbi12d 634 . . . 4 (𝑎 = 𝐴 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
3231imbi2d 344 . . 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 6669 . . . . . . 7 (𝑌‘∅) = ((rec(𝑖, ) ↾ ω)‘∅)
37 fr0g 8093 . . . . . . . 8 ( ∈ V → ((rec(𝑖, ) ↾ ω)‘∅) = )
3837elv 3403 . . . . . . 7 ((rec(𝑖, ) ↾ ω)‘∅) =
3936, 38eqtri 2761 . . . . . 6 (𝑌‘∅) =
40 f1eq1 6563 . . . . . 6 ((𝑌‘∅) = → ((𝑌‘∅):ω–1-1→V ↔ :ω–1-1→V))
4139, 40ax-mp 5 . . . . 5 ((𝑌‘∅):ω–1-1→V ↔ :ω–1-1→V)
4239rneqi 5774 . . . . . . 7 ran (𝑌‘∅) = ran
4342unieqi 4806 . . . . . 6 ran (𝑌‘∅) = ran
4443sseq1i 3903 . . . . 5 ( ran (𝑌‘∅) ⊆ 𝐺 ran 𝐺)
4541, 44anbi12i 630 . . . 4 (((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺) ↔ (:ω–1-1→V ∧ ran 𝐺))
4633, 34, 45sylanbrc 586 . . 3 (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺))
47 fin23lem.h . . . . . . . . . 10 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
48 fvex 6681 . . . . . . . . . . 11 (𝑌𝑏) ∈ V
49 f1eq1 6563 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → (𝑗:ω–1-1→V ↔ (𝑌𝑏):ω–1-1→V))
50 rneq 5773 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5150unieqd 4807 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5251sseq1d 3906 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran 𝑗𝐺 ran (𝑌𝑏) ⊆ 𝐺))
5349, 52anbi12d 634 . . . . . . . . . . . 12 (𝑗 = (𝑌𝑏) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
54 fveq2 6668 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → (𝑖𝑗) = (𝑖‘(𝑌𝑏)))
55 f1eq1 6563 . . . . . . . . . . . . . 14 ((𝑖𝑗) = (𝑖‘(𝑌𝑏)) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
5654, 55syl 17 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
5754rneqd 5775 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
5857unieqd 4807 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
5958, 51psseq12d 3983 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
6056, 59anbi12d 634 . . . . . . . . . . . 12 (𝑗 = (𝑌𝑏) → (((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗) ↔ ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏))))
6153, 60imbi12d 348 . . . . . . . . . . 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)))))
6248, 61spcv 3507 . . . . . . . . . 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))))
6347, 62syl 17 . . . . . . . . 9 (𝜑 → (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏))))
6463imp 410 . . . . . . . 8 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
65 pssss 3984 . . . . . . . . . . . 12 ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏))
66 sstr 3883 . . . . . . . . . . . 12 (( ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6765, 66sylan 583 . . . . . . . . . . 11 (( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6867expcom 417 . . . . . . . . . 10 ( ran (𝑌𝑏) ⊆ 𝐺 → ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
6968anim2d 615 . . . . . . . . 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)))
7069ad2antll 729 . . . . . . . 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)))
7164, 70mpd 15 . . . . . . 7 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
72713adant1 1131 . . . . . 6 ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
73 frsuc 8094 . . . . . . . . 9 (𝑏 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝑏) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏)))
7435fveq1i 6669 . . . . . . . . 9 (𝑌‘suc 𝑏) = ((rec(𝑖, ) ↾ ω)‘suc 𝑏)
7535fveq1i 6669 . . . . . . . . . 10 (𝑌𝑏) = ((rec(𝑖, ) ↾ ω)‘𝑏)
7675fveq2i 6671 . . . . . . . . 9 (𝑖‘(𝑌𝑏)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏))
7773, 74, 763eqtr4g 2798 . . . . . . . 8 (𝑏 ∈ ω → (𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)))
78 f1eq1 6563 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ((𝑌‘suc 𝑏):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
79 rneq 5773 . . . . . . . . . . 11 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8079unieqd 4807 . . . . . . . . . 10 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8180sseq1d 3906 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ( ran (𝑌‘suc 𝑏) ⊆ 𝐺 ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
8278, 81anbi12d 634 . . . . . . . 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)))
8377, 82syl 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)))
84833ad2ant1 1134 . . . . . 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)))
8572, 84mpbird 260 . . . . 5 ((𝑏 ∈ ω ∧ 𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))
86853exp 1120 . . . 4 (𝑏 ∈ ω → (𝜑 → (((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺) → ((𝑌‘suc 𝑏):ω–1-1→V ∧ ran (𝑌‘suc 𝑏) ⊆ 𝐺))))
8786a2d 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))))
888, 16, 24, 32, 46, 87finds 7622 . 2 (𝐴 ∈ ω → (𝜑 → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
8988impcom 411 1 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wcel 2113  {cab 2716  wral 3053  Vcvv 3397  wss 3841  wpss 3842  c0 4209  𝒫 cpw 4485   cuni 4793   cint 4833  ran crn 5520  cres 5521  suc csuc 6168  1-1wf1 6330  cfv 6333  (class class class)co 7164  ωcom 7593  reccrdg 8067  m cmap 8430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068
This theorem is referenced by:  fin23lem35  9840  fin23lem39  9843
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