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Theorem satffunlem 34392
Description: Lemma for satffunlem1lem1 34393 and satffunlem2lem1 34395. (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 2757 . . . . . . . 8 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)))
2 fvex 6905 . . . . . . . . . . . 12 (1st𝑢) ∈ V
3 fvex 6905 . . . . . . . . . . . 12 (1st𝑣) ∈ V
4 gonafv 34341 . . . . . . . . . . . 12 (((1st𝑢) ∈ V ∧ (1st𝑣) ∈ V) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
52, 3, 4mp2an 691 . . . . . . . . . . 11 ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩
6 fvex 6905 . . . . . . . . . . . 12 (1st𝑠) ∈ V
7 fvex 6905 . . . . . . . . . . . 12 (1st𝑟) ∈ V
8 gonafv 34341 . . . . . . . . . . . 12 (((1st𝑠) ∈ V ∧ (1st𝑟) ∈ V) → ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
96, 7, 8mp2an 691 . . . . . . . . . . 11 ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩
105, 9eqeq12i 2751 . . . . . . . . . 10 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
11 1oex 8476 . . . . . . . . . . 11 1o ∈ V
12 opex 5465 . . . . . . . . . . 11 ⟨(1st𝑢), (1st𝑣)⟩ ∈ V
1311, 12opth 5477 . . . . . . . . . 10 (⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ ↔ (1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩))
142, 3opth 5477 . . . . . . . . . . 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 8032 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑠𝑍) → (𝑍‘(1st𝑠)) = (2nd𝑠))
1817ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑠𝑍 → (𝑍‘(1st𝑠)) = (2nd𝑠)))
19 funfv1st2nd 8032 . . . . . . . . . . . . . . . . . . 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 8032 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑢𝑍) → (𝑍‘(1st𝑢)) = (2nd𝑢))
2322ex 414 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑢𝑍 → (𝑍‘(1st𝑢)) = (2nd𝑢)))
24 funfv1st2nd 8032 . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑠) = (1st𝑢) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2827eqcoms 2741 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑢) = (1st𝑠) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2928adantr 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
3029eqeq1d 2735 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ↔ (𝑍‘(1st𝑢)) = (2nd𝑠)))
31 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑟) = (1st𝑣) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3231eqcoms 2741 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = (1st𝑟) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3332adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3433eqeq1d 2735 . . . . . . . . . . . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . . . . . . . . . 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 4214 . . . . . . . . . . . . . . . . . . . 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 1348 . . . . . . . . . . . . . 14 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4847difeq2d 4123 . . . . . . . . . . . . 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 2750 . . . . . . . . . . . . 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 1112 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 1088   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3946  cin 3948  cop 4635  Fun wfun 6538  cfv 6544  (class class class)co 7409  ωcom 7855  1st c1st 7973  2nd c2nd 7974  1oc1o 8459  m cmap 8820  𝑔cgna 34325
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-1st 7975  df-2nd 7976  df-1o 8466  df-gona 34332
This theorem is referenced by:  satffunlem1lem1  34393  satffunlem2lem1  34395
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