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Theorem satffunlem 33995
Description: Lemma for satffunlem1lem1 33996 and satffunlem2lem1 33998. (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 2760 . . . . . . . 8 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑥 = ((1st𝑠)⊼𝑔(1st𝑟))) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)))
2 fvex 6855 . . . . . . . . . . . 12 (1st𝑢) ∈ V
3 fvex 6855 . . . . . . . . . . . 12 (1st𝑣) ∈ V
4 gonafv 33944 . . . . . . . . . . . 12 (((1st𝑢) ∈ V ∧ (1st𝑣) ∈ V) → ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩)
52, 3, 4mp2an 690 . . . . . . . . . . 11 ((1st𝑢)⊼𝑔(1st𝑣)) = ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩
6 fvex 6855 . . . . . . . . . . . 12 (1st𝑠) ∈ V
7 fvex 6855 . . . . . . . . . . . 12 (1st𝑟) ∈ V
8 gonafv 33944 . . . . . . . . . . . 12 (((1st𝑠) ∈ V ∧ (1st𝑟) ∈ V) → ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
96, 7, 8mp2an 690 . . . . . . . . . . 11 ((1st𝑠)⊼𝑔(1st𝑟)) = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩
105, 9eqeq12i 2754 . . . . . . . . . 10 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ ⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩)
11 1oex 8422 . . . . . . . . . . 11 1o ∈ V
12 opex 5421 . . . . . . . . . . 11 ⟨(1st𝑢), (1st𝑣)⟩ ∈ V
1311, 12opth 5433 . . . . . . . . . 10 (⟨1o, ⟨(1st𝑢), (1st𝑣)⟩⟩ = ⟨1o, ⟨(1st𝑠), (1st𝑟)⟩⟩ ↔ (1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩))
142, 3opth 5433 . . . . . . . . . . 11 (⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩ ↔ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)))
1514anbi2i 623 . . . . . . . . . 10 ((1o = 1o ∧ ⟨(1st𝑢), (1st𝑣)⟩ = ⟨(1st𝑠), (1st𝑟)⟩) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
1610, 13, 153bitri 296 . . . . . . . . 9 (((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑠)⊼𝑔(1st𝑟)) ↔ (1o = 1o ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))))
17 funfv1st2nd 7978 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑠𝑍) → (𝑍‘(1st𝑠)) = (2nd𝑠))
1817ex 413 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑠𝑍 → (𝑍‘(1st𝑠)) = (2nd𝑠)))
19 funfv1st2nd 7978 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑟𝑍) → (𝑍‘(1st𝑟)) = (2nd𝑟))
2019ex 413 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑟𝑍 → (𝑍‘(1st𝑟)) = (2nd𝑟)))
2118, 20anim12d 609 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑠𝑍𝑟𝑍) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟))))
22 funfv1st2nd 7978 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑢𝑍) → (𝑍‘(1st𝑢)) = (2nd𝑢))
2322ex 413 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑢𝑍 → (𝑍‘(1st𝑢)) = (2nd𝑢)))
24 funfv1st2nd 7978 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑍𝑣𝑍) → (𝑍‘(1st𝑣)) = (2nd𝑣))
2524ex 413 . . . . . . . . . . . . . . . . . 18 (Fun 𝑍 → (𝑣𝑍 → (𝑍‘(1st𝑣)) = (2nd𝑣)))
2623, 25anim12d 609 . . . . . . . . . . . . . . . . 17 (Fun 𝑍 → ((𝑢𝑍𝑣𝑍) → ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))))
27 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑠) = (1st𝑢) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2827eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑢) = (1st𝑠) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
2928adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑠)) = (𝑍‘(1st𝑢)))
3029eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑠)) = (2nd𝑠) ↔ (𝑍‘(1st𝑢)) = (2nd𝑠)))
31 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑟) = (1st𝑣) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3231eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = (1st𝑟) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3332adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑍‘(1st𝑟)) = (𝑍‘(1st𝑣)))
3433eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((𝑍‘(1st𝑟)) = (2nd𝑟) ↔ (𝑍‘(1st𝑣)) = (2nd𝑟)))
3530, 34anbi12d 631 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (((𝑍‘(1st𝑠)) = (2nd𝑠) ∧ (𝑍‘(1st𝑟)) = (2nd𝑟)) ↔ ((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟))))
3635anbi1d 630 . . . . . . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . . . . . . . . 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 4173 . . . . . . . . . . . . . . . . . . . 20 ((((𝑍‘(1st𝑢)) = (2nd𝑠) ∧ (𝑍‘(1st𝑣)) = (2nd𝑟)) ∧ ((𝑍‘(1st𝑢)) = (2nd𝑢) ∧ (𝑍‘(1st𝑣)) = (2nd𝑣))) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))
4236, 41syl6bi 252 . . . . . . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . . . . 16 (Fun 𝑍 → (((𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → ((2nd𝑠) ∩ (2nd𝑟)) = ((2nd𝑢) ∩ (2nd𝑣)))))
4645expd 416 . . . . . . . . . . . . . . 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 4082 . . . . . . . . . . . . 13 (((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) → ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))
4948adantr 481 . . . . . . . . . . . 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 2753 . . . . . . . . . . . . 13 ((𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → (𝑦 = 𝑤 ↔ ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
5150adantl 482 . . . . . . . . . . . 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 256 . . . . . . . . . . 11 ((((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ ((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟))) ∧ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
5352exp43 437 . . . . . . . . . 10 ((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) → (((1st𝑢) = (1st𝑠) ∧ (1st𝑣) = (1st𝑟)) → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟))) → (𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 = 𝑤))))
5453adantld 491 . . . . . . . . 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 416 . . . . . 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 411 . . . 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 411 . 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 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cin 3909  cop 4592  Fun wfun 6490  cfv 6496  (class class class)co 7357  ωcom 7802  1st c1st 7919  2nd c2nd 7920  1oc1o 8405  m cmap 8765  𝑔cgna 33928
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-suc 6323  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-1st 7921  df-2nd 7922  df-1o 8412  df-gona 33935
This theorem is referenced by:  satffunlem1lem1  33996  satffunlem2lem1  33998
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