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Theorem ac6sfi 8361
Description: A version of ac6s 9509 for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
Hypothesis
Ref Expression
ac6sfi.1 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
Assertion
Ref Expression
ac6sfi ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Distinct variable groups:   𝑥,𝑓,𝐴   𝑦,𝑓,𝐵,𝑥   𝜑,𝑓   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑓)   𝐴(𝑦)

Proof of Theorem ac6sfi
Dummy variables 𝑢 𝑤 𝑧 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3287 . . . 4 (𝑢 = ∅ → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑))
2 feq2 6168 . . . . . 6 (𝑢 = ∅ → (𝑓:𝑢𝐵𝑓:∅⟶𝐵))
3 raleq 3287 . . . . . 6 (𝑢 = ∅ → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ ∅ 𝜓))
42, 3anbi12d 610 . . . . 5 (𝑢 = ∅ → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
54exbidv 2002 . . . 4 (𝑢 = ∅ → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
61, 5imbi12d 333 . . 3 (𝑢 = ∅ → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))))
7 raleq 3287 . . . 4 (𝑢 = 𝑤 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝑤𝑦𝐵 𝜑))
8 feq2 6168 . . . . . 6 (𝑢 = 𝑤 → (𝑓:𝑢𝐵𝑓:𝑤𝐵))
9 raleq 3287 . . . . . 6 (𝑢 = 𝑤 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝑤 𝜓))
108, 9anbi12d 610 . . . . 5 (𝑢 = 𝑤 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
1110exbidv 2002 . . . 4 (𝑢 = 𝑤 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
127, 11imbi12d 333 . . 3 (𝑢 = 𝑤 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓))))
13 raleq 3287 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑))
14 feq2 6168 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (𝑓:𝑢𝐵𝑓:(𝑤 ∪ {𝑧})⟶𝐵))
15 raleq 3287 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓))
1614, 15anbi12d 610 . . . . . 6 (𝑢 = (𝑤 ∪ {𝑧}) → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
1716exbidv 2002 . . . . 5 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18 feq1 6167 . . . . . . 7 (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19 fvex 6343 . . . . . . . . . 10 (𝑓𝑥) ∈ V
20 ac6sfi.1 . . . . . . . . . 10 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
2119, 20sbcie 3623 . . . . . . . . 9 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
22 fveq1 6332 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
2322sbceq1d 3593 . . . . . . . . 9 (𝑓 = 𝑔 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔𝑥) / 𝑦]𝜑))
2421, 23syl5bbr 274 . . . . . . . 8 (𝑓 = 𝑔 → (𝜓[(𝑔𝑥) / 𝑦]𝜑))
2524ralbidv 3135 . . . . . . 7 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2618, 25anbi12d 610 . . . . . 6 (𝑓 = 𝑔 → ((𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2726cbvexvw 2126 . . . . 5 (∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2817, 27syl6bb 276 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2913, 28imbi12d 333 . . 3 (𝑢 = (𝑤 ∪ {𝑧}) → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
30 raleq 3287 . . . 4 (𝑢 = 𝐴 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜑))
31 feq2 6168 . . . . . 6 (𝑢 = 𝐴 → (𝑓:𝑢𝐵𝑓:𝐴𝐵))
32 raleq 3287 . . . . . 6 (𝑢 = 𝐴 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝐴 𝜓))
3331, 32anbi12d 610 . . . . 5 (𝑢 = 𝐴 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3433exbidv 2002 . . . 4 (𝑢 = 𝐴 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3530, 34imbi12d 333 . . 3 (𝑢 = 𝐴 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))))
36 f0 6227 . . . 4 ∅:∅⟶𝐵
37 0ex 4925 . . . . 5 ∅ ∈ V
38 ral0 4218 . . . . . . 7 𝑥 ∈ ∅ 𝜓
3938biantru 515 . . . . . 6 (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40 feq1 6167 . . . . . 6 (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
4139, 40syl5bbr 274 . . . . 5 (𝑓 = ∅ → ((𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓) ↔ ∅:∅⟶𝐵))
4237, 41spcev 3452 . . . 4 (∅:∅⟶𝐵 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
4336, 42mp1i 13 . . 3 (∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
44 ssun1 3928 . . . . . . 7 𝑤 ⊆ (𝑤 ∪ {𝑧})
45 ssralv 3816 . . . . . . 7 (𝑤 ⊆ (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∀𝑥𝑤𝑦𝐵 𝜑))
4644, 45ax-mp 5 . . . . . 6 (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∀𝑥𝑤𝑦𝐵 𝜑)
4746imim1i 63 . . . . 5 ((∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
48 ssun2 3929 . . . . . . . . 9 {𝑧} ⊆ (𝑤 ∪ {𝑧})
49 ssralv 3816 . . . . . . . . 9 ({𝑧} ⊆ (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑))
5048, 49ax-mp 5 . . . . . . . 8 (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑)
51 vex 3354 . . . . . . . . . 10 𝑧 ∈ V
52 ralsnsg 4355 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑))
5351, 52ax-mp 5 . . . . . . . . 9 (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
54 sbcrex 3664 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5553, 54bitri 264 . . . . . . . 8 (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5650, 55sylib 208 . . . . . . 7 (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
57 nfv 1995 . . . . . . . 8 𝑦 ¬ 𝑧𝑤
58 nfv 1995 . . . . . . . . 9 𝑦𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)
59 nfv 1995 . . . . . . . . . . 11 𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
60 nfcv 2913 . . . . . . . . . . . 12 𝑦(𝑤 ∪ {𝑧})
61 nfsbc1v 3608 . . . . . . . . . . . 12 𝑦[(𝑔𝑥) / 𝑦]𝜑
6260, 61nfral 3094 . . . . . . . . . . 11 𝑦𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑
6359, 62nfan 1980 . . . . . . . . . 10 𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6463nfex 2318 . . . . . . . . 9 𝑦𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6558, 64nfim 1977 . . . . . . . 8 𝑦(∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
66 simprl 748 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑓:𝑤𝐵)
67 vex 3354 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
6851, 67f1osn 6318 . . . . . . . . . . . . . . 15 {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69 f1of 6279 . . . . . . . . . . . . . . 15 ({⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦} → {⟨𝑧, 𝑦⟩}:{𝑧}⟶{𝑦})
7068, 69mp1i 13 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶{𝑦})
71 simpl2 1229 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦𝐵)
7271snssd 4476 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
7370, 72fssd 6198 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵)
74 simpl1 1227 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ¬ 𝑧𝑤)
75 disjsn 4384 . . . . . . . . . . . . . 14 ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑤)
7674, 75sylibr 224 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑤 ∩ {𝑧}) = ∅)
77 fun2 6208 . . . . . . . . . . . . 13 (((𝑓:𝑤𝐵 ∧ {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵) ∧ (𝑤 ∩ {𝑧}) = ∅) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
7866, 73, 76, 77syl21anc 1475 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
79 simprr 750 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 𝜓)
80 eleq1a 2845 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑤 → (𝑧 = 𝑥𝑧𝑤))
8180necon3bd 2957 . . . . . . . . . . . . . . . . . 18 (𝑥𝑤 → (¬ 𝑧𝑤𝑧𝑥))
8281impcom 394 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑤𝑥𝑤) → 𝑧𝑥)
83 fvunsn 6590 . . . . . . . . . . . . . . . . 17 (𝑧𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥))
84 dfsbcq 3590 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥) → ([((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
8584, 21syl6rbb 277 . . . . . . . . . . . . . . . . 17 (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥) → (𝜓[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8682, 83, 853syl 18 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑤𝑥𝑤) → (𝜓[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8786ralbidva 3134 . . . . . . . . . . . . . . 15 𝑧𝑤 → (∀𝑥𝑤 𝜓 ↔ ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8874, 87syl 17 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥𝑤 𝜓 ↔ ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8979, 88mpbid 222 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
90 simpl3 1231 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → [𝑧 / 𝑥]𝜑)
91 ffun 6189 . . . . . . . . . . . . . . . . 17 ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
92 ssun2 3929 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
93 vsnid 4349 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
9467dmsnop 5752 . . . . . . . . . . . . . . . . . . 19 dom {⟨𝑧, 𝑦⟩} = {𝑧}
9593, 94eleqtrri 2849 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
96 funssfv 6351 . . . . . . . . . . . . . . . . . 18 ((Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9792, 95, 96mp3an23 1564 . . . . . . . . . . . . . . . . 17 (Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9878, 91, 973syl 18 . . . . . . . . . . . . . . . 16 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9951, 67fvsn 6591 . . . . . . . . . . . . . . . 16 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
10098, 99syl6req 2822 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
101 ralsnsg 4355 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑))
10251, 101ax-mp 5 . . . . . . . . . . . . . . . 16 (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑)
103 elsni 4334 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
104103fveq2d 6337 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
105104eqeq2d 2781 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑧} → (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) ↔ 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧)))
106105biimparc 465 . . . . . . . . . . . . . . . . . 18 ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
107 sbceq1a 3599 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) → (𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
108106, 107syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → (𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
109108ralbidva 3134 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → (∀𝑥 ∈ {𝑧}𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
110102, 109syl5bbr 274 . . . . . . . . . . . . . . 15 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
111100, 110syl 17 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
11290, 111mpbid 222 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
113 ralun 3947 . . . . . . . . . . . . 13 ((∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑 ∧ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
11489, 112, 113syl2anc 567 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
115 vex 3354 . . . . . . . . . . . . . 14 𝑓 ∈ V
116 snex 5037 . . . . . . . . . . . . . 14 {⟨𝑧, 𝑦⟩} ∈ V
117115, 116unex 7104 . . . . . . . . . . . . 13 (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
118 feq1 6167 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
119 fveq1 6332 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
120119sbceq1d 3593 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔𝑥) / 𝑦]𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
121120ralbidv 3135 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
122118, 121anbi12d 610 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ((𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑) ↔ ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)))
123117, 122spcev 3452 . . . . . . . . . . . 12 (((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
12478, 114, 123syl2anc 567 . . . . . . . . . . 11 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
125124ex 397 . . . . . . . . . 10 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → ((𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
126125exlimdv 2013 . . . . . . . . 9 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
1271263exp 1112 . . . . . . . 8 𝑧𝑤 → (𝑦𝐵 → ([𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))))
12857, 65, 127rexlimd 3174 . . . . . . 7 𝑧𝑤 → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
12956, 128syl5 34 . . . . . 6 𝑧𝑤 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
130129a2d 29 . . . . 5 𝑧𝑤 → ((∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
13147, 130syl5 34 . . . 4 𝑧𝑤 → ((∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
132131adantl 467 . . 3 ((𝑤 ∈ Fin ∧ ¬ 𝑧𝑤) → ((∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
1336, 12, 29, 35, 43, 132findcard2s 8358 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
134133imp 393 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  [wsbc 3588  cun 3722  cin 3723  wss 3724  c0 4064  {csn 4317  cop 4323  dom cdm 5250  Fun wfun 6026  wf 6028  1-1-ontowf1o 6031  cfv 6032  Fincfn 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-om 7214  df-1o 7714  df-er 7897  df-en 8111  df-fin 8114
This theorem is referenced by:  fissuni  8428  fipreima  8429  indexfi  8431  finacn  9074  axcc4dom  9466  ttukeylem6  9539  firest  16302  ablfaclem3  18695  ablfac2  18697  cmpcovf  21416  cmpsub  21425  tgcmp  21426  hauscmplem  21431  comppfsc  21557  ptcnplem  21646  alexsubALTlem3  22074  alexsubALT  22076  tsmsxplem1  22177  ovolicc2lem5  23510  ovolicc2  23511  limciun  23879  cvmliftlem15  31619  matunitlindflem2  33740  ptrecube  33743  istotbnd3  33903  sstotbnd2  33906  sstotbnd  33907  prdsbnd  33925  prdstotbnd  33926  heiborlem1  33943  heibor  33953  kelac1  38160  hbt  38227
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