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Theorem fin23lem34 10289
Description: Lemma for fin23 10332. 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 6847 . . . . . 6 (𝑎 = ∅ → (𝑌𝑎) = (𝑌‘∅))
2 f1eq1 6738 . . . . . 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 5898 . . . . . . 7 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
54unieqd 4884 . . . . . 6 (𝑎 = ∅ → ran (𝑌𝑎) = ran (𝑌‘∅))
65sseq1d 3980 . . . . 5 (𝑎 = ∅ → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘∅) ⊆ 𝐺))
73, 6anbi12d 632 . . . 4 (𝑎 = ∅ → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺)))
87imbi2d 341 . . 3 (𝑎 = ∅ → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺))))
9 fveq2 6847 . . . . . 6 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
10 f1eq1 6738 . . . . . 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 5898 . . . . . . 7 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
1312unieqd 4884 . . . . . 6 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
1413sseq1d 3980 . . . . 5 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝑏) ⊆ 𝐺))
1511, 14anbi12d 632 . . . 4 (𝑎 = 𝑏 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
1615imbi2d 341 . . 3 (𝑎 = 𝑏 → ((𝜑 → ((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺)) ↔ (𝜑 → ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺))))
17 fveq2 6847 . . . . . 6 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
18 f1eq1 6738 . . . . . 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 5898 . . . . . . 7 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2120unieqd 4884 . . . . . 6 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
2221sseq1d 3980 . . . . 5 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌‘suc 𝑏) ⊆ 𝐺))
2319, 22anbi12d 632 . . . 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 341 . . 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 6847 . . . . . 6 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
26 f1eq1 6738 . . . . . 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 5898 . . . . . . 7 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
2928unieqd 4884 . . . . . 6 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
3029sseq1d 3980 . . . . 5 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ 𝐺 ran (𝑌𝐴) ⊆ 𝐺))
3127, 30anbi12d 632 . . . 4 (𝑎 = 𝐴 → (((𝑌𝑎):ω–1-1→V ∧ ran (𝑌𝑎) ⊆ 𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
3231imbi2d 341 . . 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 6848 . . . . . . 7 (𝑌‘∅) = ((rec(𝑖, ) ↾ ω)‘∅)
37 fr0g 8387 . . . . . . . 8 ( ∈ V → ((rec(𝑖, ) ↾ ω)‘∅) = )
3837elv 3454 . . . . . . 7 ((rec(𝑖, ) ↾ ω)‘∅) =
3936, 38eqtri 2765 . . . . . 6 (𝑌‘∅) =
40 f1eq1 6738 . . . . . 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 5897 . . . . . . 7 ran (𝑌‘∅) = ran
4342unieqi 4883 . . . . . 6 ran (𝑌‘∅) = ran
4443sseq1i 3977 . . . . 5 ( ran (𝑌‘∅) ⊆ 𝐺 ran 𝐺)
4541, 44anbi12i 628 . . . 4 (((𝑌‘∅):ω–1-1→V ∧ ran (𝑌‘∅) ⊆ 𝐺) ↔ (:ω–1-1→V ∧ ran 𝐺))
4633, 34, 45sylanbrc 584 . . 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 6860 . . . . . . . . . . 11 (𝑌𝑏) ∈ V
49 f1eq1 6738 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → (𝑗:ω–1-1→V ↔ (𝑌𝑏):ω–1-1→V))
50 rneq 5896 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5150unieqd 4884 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran 𝑗 = ran (𝑌𝑏))
5251sseq1d 3980 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran 𝑗𝐺 ran (𝑌𝑏) ⊆ 𝐺))
5349, 52anbi12d 632 . . . . . . . . . . . 12 (𝑗 = (𝑌𝑏) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)))
54 fveq2 6847 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → (𝑖𝑗) = (𝑖‘(𝑌𝑏)))
55 f1eq1 6738 . . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . 15 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
5857unieqd 4884 . . . . . . . . . . . . . 14 (𝑗 = (𝑌𝑏) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝑏)))
5958, 51psseq12d 4059 . . . . . . . . . . . . 13 (𝑗 = (𝑌𝑏) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
6056, 59anbi12d 632 . . . . . . . . . . . 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 345 . . . . . . . . . . 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 3567 . . . . . . . . . 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 408 . . . . . . . 8 ((𝜑 ∧ ((𝑌𝑏):ω–1-1→V ∧ ran (𝑌𝑏) ⊆ 𝐺)) → ((𝑖‘(𝑌𝑏)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏)))
65 pssss 4060 . . . . . . . . . . . 12 ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏))
66 sstr 3957 . . . . . . . . . . . 12 (( ran (𝑖‘(𝑌𝑏)) ⊆ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6765, 66sylan 581 . . . . . . . . . . 11 (( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) ∧ ran (𝑌𝑏) ⊆ 𝐺) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺)
6867expcom 415 . . . . . . . . . 10 ( ran (𝑌𝑏) ⊆ 𝐺 → ( ran (𝑖‘(𝑌𝑏)) ⊊ ran (𝑌𝑏) → ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
6968anim2d 613 . . . . . . . . 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 728 . . . . . . . 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 8388 . . . . . . . . 9 (𝑏 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝑏) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏)))
7435fveq1i 6848 . . . . . . . . 9 (𝑌‘suc 𝑏) = ((rec(𝑖, ) ↾ ω)‘suc 𝑏)
7535fveq1i 6848 . . . . . . . . . 10 (𝑌𝑏) = ((rec(𝑖, ) ↾ ω)‘𝑏)
7675fveq2i 6850 . . . . . . . . 9 (𝑖‘(𝑌𝑏)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝑏))
7773, 74, 763eqtr4g 2802 . . . . . . . 8 (𝑏 ∈ ω → (𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)))
78 f1eq1 6738 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ((𝑌‘suc 𝑏):ω–1-1→V ↔ (𝑖‘(𝑌𝑏)):ω–1-1→V))
79 rneq 5896 . . . . . . . . . . 11 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8079unieqd 4884 . . . . . . . . . 10 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ran (𝑌‘suc 𝑏) = ran (𝑖‘(𝑌𝑏)))
8180sseq1d 3980 . . . . . . . . 9 ((𝑌‘suc 𝑏) = (𝑖‘(𝑌𝑏)) → ( ran (𝑌‘suc 𝑏) ⊆ 𝐺 ran (𝑖‘(𝑌𝑏)) ⊆ 𝐺))
8278, 81anbi12d 632 . . . . . . . 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 257 . . . . 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 7840 . 2 (𝐴 ∈ ω → (𝜑 → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
8988impcom 409 1 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  {cab 2714  wral 3065  Vcvv 3448  wss 3915  wpss 3916  c0 4287  𝒫 cpw 4565   cuni 4870   cint 4912  ran crn 5639  cres 5640  suc csuc 6324  1-1wf1 6498  cfv 6501  (class class class)co 7362  ωcom 7807  reccrdg 8360  m cmap 8772
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361
This theorem is referenced by:  fin23lem35  10290  fin23lem39  10293
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