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Theorem satffunlem 33726
Description: Lemma for satffunlem1lem1 33727 and satffunlem2lem1 33729. (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 2761 . . . . . . . 8 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)))
2 fvex 6847 . . . . . . . . . . . 12 (1st𝑢) ∈ V
3 fvex 6847 . . . . . . . . . . . 12 (1st𝑣) ∈ V
4 gonafv 33675 . . . . . . . . . . . 12 (((1st𝑢) ∈ V ∧ (1st𝑣) ∈ V) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
52, 3, 4mp2an 690 . . . . . . . . . . 11 ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩
6 fvex 6847 . . . . . . . . . . . 12 (1st𝑠) ∈ V
7 fvex 6847 . . . . . . . . . . . 12 (1st𝑟) ∈ V
8 gonafv 33675 . . . . . . . . . . . 12 (((1st𝑠) ∈ V ∧ (1st𝑟) ∈ V) → ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
96, 7, 8mp2an 690 . . . . . . . . . . 11 ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩
105, 9eqeq12i 2755 . . . . . . . . . 10 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
11 1oex 8386 . . . . . . . . . . 11 1o ∈ V
12 opex 5416 . . . . . . . . . . 11 ⟨(1st𝑢), (1st𝑣)⟩ ∈ V
1311, 12opth 5428 . . . . . . . . . 10 (⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ ↔ (1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩))
142, 3opth 5428 . . . . . . . . . . 11 (⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩ ↔ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)))
1514anbi2i 624 . . . . . . . . . 10 ((1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
1610, 13, 153bitri 297 . . . . . . . . 9 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
17 funfv1st2nd 7964 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑠𝑍) → (𝑍‘(1st𝑠)) = (2nd𝑠))
1817ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑠𝑍 → (𝑍‘(1st𝑠)) = (2nd𝑠)))
19 funfv1st2nd 7964 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑟𝑍) → (𝑍‘(1st𝑟)) = (2nd𝑟))
2019ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑟𝑍 → (𝑍‘(1st𝑟)) = (2nd𝑟)))
2118, 20anim12d 610 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑠𝑍𝑟𝑍) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟))))
22 funfv1st2nd 7964 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑢𝑍) → (𝑍‘(1st𝑢)) = (2nd𝑢))
2322ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑢𝑍 → (𝑍‘(1st𝑢)) = (2nd𝑢)))
24 funfv1st2nd 7964 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑣𝑍) → (𝑍‘(1st𝑣)) = (2nd𝑣))
2524ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑣𝑍 → (𝑍‘(1st𝑣)) = (2nd𝑣)))
2623, 25anim12d 610 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑢𝑍𝑣𝑍) → ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))))
27 fveq2 6834 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑠) = (1st𝑢) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2827eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑢) = (1st𝑠) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2928adantr 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
3029eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ↔ (𝑍‘(1st𝑢)) = (2nd𝑠)))
31 fveq2 6834 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑟) = (1st𝑣) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3231eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = (1st𝑟) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3332adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3433eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑟)) = (2nd𝑟) ↔ (𝑍‘(1st𝑣)) = (2nd𝑟)))
3530, 34anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ↔ ((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟))))
3635anbi1d 631 . . . . . . . . . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑢)) = (2nd𝑢)) → (2nd𝑠) = (2nd𝑢))
3837ad2ant2r 745 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (2nd𝑠) = (2nd𝑢))
39 eqtr2 2761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑍‘(1st𝑣)) = (2nd𝑟) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣)) → (2nd𝑟) = (2nd𝑣))
4039ad2ant2l 744 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → (2nd𝑟) = (2nd𝑣))
4138, 40ineq12d 4168 . . . . . . . . . . . . . . . . . . . 20 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4236, 41syl6bi 253 . . . . . . . . . . . . . . . . . . 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 609 . . . . . . . . . . . . . . . 16 (Fun 𝑍 → (((𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))))
4645expd 417 . . . . . . . . . . . . . . 15 (Fun 𝑍 → ((𝑠𝑍𝑟𝑍) → ((𝑢𝑍𝑣𝑍) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣))))))
47463imp1 1347 . . . . . . . . . . . . . 14 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4847difeq2d 4077 . . . . . . . . . . . . 13 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))
4948adantr 482 . . . . . . . . . . . 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 2754 . . . . . . . . . . . . 13 ((𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → (𝑦 = 𝑤 ↔ ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
5150adantl 483 . . . . . . . . . . . 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 257 . . . . . . . . . . 11 ((((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) ∧ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
5352exp43 438 . . . . . . . . . 10 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5453adantld 492 . . . . . . . . 9 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → ((1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5516, 54biimtrid 241 . . . . . . . 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 417 . . . . . 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 412 . . . 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 412 . 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 1111 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 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3443  cdif 3902  cin 3904  cop 4587  Fun wfun 6482  cfv 6488  (class class class)co 7346  ωcom 7789  1st c1st 7906  2nd c2nd 7907  1oc1o 8369  m cmap 8695  𝑔cgna 33659
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379  ax-un 7659
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-suc 6316  df-iota 6440  df-fun 6490  df-fv 6496  df-ov 7349  df-1st 7908  df-2nd 7909  df-1o 8376  df-gona 33666
This theorem is referenced by:  satffunlem1lem1  33727  satffunlem2lem1  33729
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