Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  satffunlem Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator