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Theorem satffunlem 32758
Description: Lemma for satffunlem1lem1 32759 and satffunlem2lem1 32761. (Contributed by AV, 27-Oct-2023.)
Assertion
Ref Expression
satffunlem (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)

Proof of Theorem satffunlem
StepHypRef Expression
1 eqtr2 2819 . . . . . . . 8 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)))
2 fvex 6658 . . . . . . . . . . . 12 (1st𝑢) ∈ V
3 fvex 6658 . . . . . . . . . . . 12 (1st𝑣) ∈ V
4 gonafv 32707 . . . . . . . . . . . 12 (((1st𝑢) ∈ V ∧ (1st𝑣) ∈ V) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
52, 3, 4mp2an 691 . . . . . . . . . . 11 ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩
6 fvex 6658 . . . . . . . . . . . 12 (1st𝑠) ∈ V
7 fvex 6658 . . . . . . . . . . . 12 (1st𝑟) ∈ V
8 gonafv 32707 . . . . . . . . . . . 12 (((1st𝑠) ∈ V ∧ (1st𝑟) ∈ V) → ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
96, 7, 8mp2an 691 . . . . . . . . . . 11 ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩
105, 9eqeq12i 2813 . . . . . . . . . 10 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
11 1oex 8093 . . . . . . . . . . 11 1o ∈ V
12 opex 5321 . . . . . . . . . . 11 ⟨(1st𝑢), (1st𝑣)⟩ ∈ V
1311, 12opth 5333 . . . . . . . . . 10 (⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ ↔ (1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩))
142, 3opth 5333 . . . . . . . . . . 11 (⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩ ↔ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)))
1514anbi2i 625 . . . . . . . . . 10 ((1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
1610, 13, 153bitri 300 . . . . . . . . 9 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
17 funfv1st2nd 7727 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑠𝑍) → (𝑍‘(1st𝑠)) = (2nd𝑠))
1817ex 416 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑠𝑍 → (𝑍‘(1st𝑠)) = (2nd𝑠)))
19 funfv1st2nd 7727 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑟𝑍) → (𝑍‘(1st𝑟)) = (2nd𝑟))
2019ex 416 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑟𝑍 → (𝑍‘(1st𝑟)) = (2nd𝑟)))
2118, 20anim12d 611 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑠𝑍𝑟𝑍) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟))))
22 funfv1st2nd 7727 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑢𝑍) → (𝑍‘(1st𝑢)) = (2nd𝑢))
2322ex 416 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑢𝑍 → (𝑍‘(1st𝑢)) = (2nd𝑢)))
24 funfv1st2nd 7727 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑣𝑍) → (𝑍‘(1st𝑣)) = (2nd𝑣))
2524ex 416 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑣𝑍 → (𝑍‘(1st𝑣)) = (2nd𝑣)))
2623, 25anim12d 611 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑢𝑍𝑣𝑍) → ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))))
27 fveq2 6645 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑠) = (1st𝑢) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2827eqcoms 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑢) = (1st𝑠) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2928adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
3029eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ↔ (𝑍‘(1st𝑢)) = (2nd𝑠)))
31 fveq2 6645 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑟) = (1st𝑣) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3231eqcoms 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = (1st𝑟) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3332adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3433eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑟)) = (2nd𝑟) ↔ (𝑍‘(1st𝑣)) = (2nd𝑟)))
3530, 34anbi12d 633 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ↔ ((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟))))
3635anbi1d 632 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) ↔ (((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣)))))
37 eqtr2 2819 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑢)) = (2nd𝑢)) → (2nd𝑠) = (2nd𝑢))
3837ad2ant2r 746 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (2nd𝑠) = (2nd𝑢))
39 eqtr2 2819 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑍‘(1st𝑣)) = (2nd𝑟) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣)) → (2nd𝑟) = (2nd𝑣))
4039ad2ant2l 745 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (2nd𝑟) = (2nd𝑣))
4138, 40ineq12d 4140 . . . . . . . . . . . . . . . . . . . 20 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4236, 41syl6bi 256 . . . . . . . . . . . . . . . . . . 19 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣))))
4342com12 32 . . . . . . . . . . . . . . . . . 18 ((((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣))))
4443a1i 11 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))))
4521, 26, 44syl2and 610 . . . . . . . . . . . . . . . 16 (Fun 𝑍 → (((𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))))
4645expd 419 . . . . . . . . . . . . . . 15 (Fun 𝑍 → ((𝑠𝑍𝑟𝑍) → ((𝑢𝑍𝑣𝑍) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣))))))
47463imp1 1344 . . . . . . . . . . . . . 14 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4847difeq2d 4050 . . . . . . . . . . . . 13 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))
4948adantr 484 . . . . . . . . . . . 12 ((((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) ∧ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))
50 eqeq12 2812 . . . . . . . . . . . . 13 ((𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → (𝑦 = 𝑤 ↔ ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
5150adantl 485 . . . . . . . . . . . 12 ((((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) ∧ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → (𝑦 = 𝑤 ↔ ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
5249, 51mpbird 260 . . . . . . . . . . 11 ((((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) ∧ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
5352exp43 440 . . . . . . . . . 10 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5453adantld 494 . . . . . . . . 9 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → ((1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5516, 54syl5bi 245 . . . . . . . 8 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
561, 55syl5 34 . . . . . . 7 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5756expd 419 . . . . . 6 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤)))))
5857com35 98 . . . . 5 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) → 𝑦 = 𝑤)))))
5958impd 414 . . . 4 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) → 𝑦 = 𝑤))))
6059com24 95 . . 3 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → 𝑦 = 𝑤))))
6160impd 414 . 2 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → ((𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → 𝑦 = 𝑤)))
62613imp 1108 1 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  Vcvv 3441  cdif 3878  cin 3880  cop 4531  Fun wfun 6318  cfv 6324  (class class class)co 7135  ωcom 7560  1st c1st 7669  2nd c2nd 7670  1oc1o 8078  m cmap 8389  𝑔cgna 32691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  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-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-1st 7671  df-2nd 7672  df-1o 8085  df-gona 32698
This theorem is referenced by:  satffunlem1lem1  32759  satffunlem2lem1  32761
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