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Theorem ac6sfi 8751
Description: A version of ac6s 9895 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 3411 . . . 4 (𝑢 = ∅ → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑))
2 feq2 6493 . . . . . 6 (𝑢 = ∅ → (𝑓:𝑢𝐵𝑓:∅⟶𝐵))
3 raleq 3411 . . . . . 6 (𝑢 = ∅ → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ ∅ 𝜓))
42, 3anbi12d 630 . . . . 5 (𝑢 = ∅ → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
54exbidv 1915 . . . 4 (𝑢 = ∅ → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
61, 5imbi12d 346 . . 3 (𝑢 = ∅ → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))))
7 raleq 3411 . . . 4 (𝑢 = 𝑤 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝑤𝑦𝐵 𝜑))
8 feq2 6493 . . . . . 6 (𝑢 = 𝑤 → (𝑓:𝑢𝐵𝑓:𝑤𝐵))
9 raleq 3411 . . . . . 6 (𝑢 = 𝑤 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝑤 𝜓))
108, 9anbi12d 630 . . . . 5 (𝑢 = 𝑤 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
1110exbidv 1915 . . . 4 (𝑢 = 𝑤 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
127, 11imbi12d 346 . . 3 (𝑢 = 𝑤 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓))))
13 raleq 3411 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑))
14 feq2 6493 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (𝑓:𝑢𝐵𝑓:(𝑤 ∪ {𝑧})⟶𝐵))
15 raleq 3411 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓))
1614, 15anbi12d 630 . . . . . 6 (𝑢 = (𝑤 ∪ {𝑧}) → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
1716exbidv 1915 . . . . 5 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18 feq1 6492 . . . . . . 7 (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19 fvex 6680 . . . . . . . . . 10 (𝑓𝑥) ∈ V
20 ac6sfi.1 . . . . . . . . . 10 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
2119, 20sbcie 3816 . . . . . . . . 9 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
22 fveq1 6666 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
2322sbceq1d 3781 . . . . . . . . 9 (𝑓 = 𝑔 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔𝑥) / 𝑦]𝜑))
2421, 23syl5bbr 286 . . . . . . . 8 (𝑓 = 𝑔 → (𝜓[(𝑔𝑥) / 𝑦]𝜑))
2524ralbidv 3202 . . . . . . 7 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2618, 25anbi12d 630 . . . . . 6 (𝑓 = 𝑔 → ((𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2726cbvexvw 2037 . . . . 5 (∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2817, 27syl6bb 288 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2913, 28imbi12d 346 . . 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 3411 . . . 4 (𝑢 = 𝐴 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜑))
31 feq2 6493 . . . . . 6 (𝑢 = 𝐴 → (𝑓:𝑢𝐵𝑓:𝐴𝐵))
32 raleq 3411 . . . . . 6 (𝑢 = 𝐴 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝐴 𝜓))
3331, 32anbi12d 630 . . . . 5 (𝑢 = 𝐴 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3433exbidv 1915 . . . 4 (𝑢 = 𝐴 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3530, 34imbi12d 346 . . 3 (𝑢 = 𝐴 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))))
36 f0 6557 . . . 4 ∅:∅⟶𝐵
37 0ex 5208 . . . . 5 ∅ ∈ V
38 ral0 4459 . . . . . . 7 𝑥 ∈ ∅ 𝜓
3938biantru 530 . . . . . 6 (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40 feq1 6492 . . . . . 6 (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
4139, 40syl5bbr 286 . . . . 5 (𝑓 = ∅ → ((𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓) ↔ ∅:∅⟶𝐵))
4237, 41spcev 3611 . . . 4 (∅:∅⟶𝐵 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
4336, 42mp1i 13 . . 3 (∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
44 ssun1 4152 . . . . . . 7 𝑤 ⊆ (𝑤 ∪ {𝑧})
45 ssralv 4037 . . . . . . 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 4153 . . . . . . . . 9 {𝑧} ⊆ (𝑤 ∪ {𝑧})
49 ssralv 4037 . . . . . . . . 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 ralsnsg 4605 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑))
5251elv 3505 . . . . . . . . 9 (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
53 sbcrex 3862 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5452, 53bitri 276 . . . . . . . 8 (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5550, 54sylib 219 . . . . . . 7 (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
56 nfv 1908 . . . . . . . 8 𝑦 ¬ 𝑧𝑤
57 nfv 1908 . . . . . . . . 9 𝑦𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)
58 nfv 1908 . . . . . . . . . . 11 𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
59 nfcv 2982 . . . . . . . . . . . 12 𝑦(𝑤 ∪ {𝑧})
60 nfsbc1v 3796 . . . . . . . . . . . 12 𝑦[(𝑔𝑥) / 𝑦]𝜑
6159, 60nfralw 3230 . . . . . . . . . . 11 𝑦𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑
6258, 61nfan 1893 . . . . . . . . . 10 𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6362nfex 2337 . . . . . . . . 9 𝑦𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6457, 63nfim 1890 . . . . . . . 8 𝑦(∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
65 simprl 767 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑓:𝑤𝐵)
66 vex 3503 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
67 vex 3503 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
6866, 67f1osn 6651 . . . . . . . . . . . . . . 15 {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69 f1of 6612 . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦𝐵)
7271snssd 4741 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
7370, 72fssd 6525 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵)
74 simpl1 1185 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ¬ 𝑧𝑤)
75 disjsn 4646 . . . . . . . . . . . . . 14 ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑤)
7674, 75sylibr 235 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑤 ∩ {𝑧}) = ∅)
7765, 73, 76fun2d 6539 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
78 simprr 769 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 𝜓)
79 eleq1a 2913 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑤 → (𝑧 = 𝑥𝑧𝑤))
8079necon3bd 3035 . . . . . . . . . . . . . . . . . 18 (𝑥𝑤 → (¬ 𝑧𝑤𝑧𝑥))
8180impcom 408 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑤𝑥𝑤) → 𝑧𝑥)
82 fvunsn 6937 . . . . . . . . . . . . . . . . 17 (𝑧𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥))
83 dfsbcq 3778 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥) → ([((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
8483, 21syl6rbb 289 . . . . . . . . . . . . . . . . 17 (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥) → (𝜓[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8581, 82, 843syl 18 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑤𝑥𝑤) → (𝜓[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8685ralbidva 3201 . . . . . . . . . . . . . . 15 𝑧𝑤 → (∀𝑥𝑤 𝜓 ↔ ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8774, 86syl 17 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (∀𝑥𝑤 𝜓 ↔ ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
8878, 87mpbid 233 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
89 simpl3 1187 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → [𝑧 / 𝑥]𝜑)
90 ffun 6514 . . . . . . . . . . . . . . . . 17 ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
91 ssun2 4153 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
92 vsnid 4599 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
9367dmsnop 6071 . . . . . . . . . . . . . . . . . . 19 dom {⟨𝑧, 𝑦⟩} = {𝑧}
9492, 93eleqtrri 2917 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
95 funssfv 6688 . . . . . . . . . . . . . . . . . 18 ((Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9691, 94, 95mp3an23 1446 . . . . . . . . . . . . . . . . 17 (Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9777, 90, 963syl 18 . . . . . . . . . . . . . . . 16 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9866, 67fvsn 6939 . . . . . . . . . . . . . . . 16 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
9997, 98syl6req 2878 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
100 ralsnsg 4605 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑))
101100elv 3505 . . . . . . . . . . . . . . . 16 (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑)
102 elsni 4581 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
103102fveq2d 6671 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
104103eqeq2d 2837 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑧} → (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) ↔ 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧)))
105104biimparc 480 . . . . . . . . . . . . . . . . . 18 ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
106 sbceq1a 3787 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) → (𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
107105, 106syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → (𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
108107ralbidva 3201 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → (∀𝑥 ∈ {𝑧}𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
109101, 108syl5bbr 286 . . . . . . . . . . . . . . 15 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
11099, 109syl 17 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
11189, 110mpbid 233 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
112 ralun 4172 . . . . . . . . . . . . 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)
11388, 111, 112syl2anc 584 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
114 vex 3503 . . . . . . . . . . . . . 14 𝑓 ∈ V
115 snex 5328 . . . . . . . . . . . . . 14 {⟨𝑧, 𝑦⟩} ∈ V
116114, 115unex 7458 . . . . . . . . . . . . 13 (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
117 feq1 6492 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
118 fveq1 6666 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
119118sbceq1d 3781 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔𝑥) / 𝑦]𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
120119ralbidv 3202 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
121117, 120anbi12d 630 . . . . . . . . . . . . 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)))
122116, 121spcev 3611 . . . . . . . . . . . 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))
12377, 113, 122syl2anc 584 . . . . . . . . . . 11 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
124123ex 413 . . . . . . . . . 10 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → ((𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
125124exlimdv 1927 . . . . . . . . 9 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
1261253exp 1113 . . . . . . . 8 𝑧𝑤 → (𝑦𝐵 → ([𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))))
12756, 64, 126rexlimd 3322 . . . . . . 7 𝑧𝑤 → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
12855, 127syl5 34 . . . . . 6 𝑧𝑤 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))))
129128a2d 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))))
13047, 129syl5 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))))
131130adantl 482 . . 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))))
1326, 12, 29, 35, 43, 131findcard2s 8748 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
133132imp 407 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  [wsbc 3776  cun 3938  cin 3939  wss 3940  c0 4295  {csn 4564  cop 4570  dom cdm 5554  Fun wfun 6346  wf 6348  1-1-ontowf1o 6351  cfv 6352  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1o 8093  df-er 8279  df-en 8499  df-fin 8502
This theorem is referenced by:  fissuni  8818  fipreima  8819  indexfi  8821  finacn  9465  axcc4dom  9852  ttukeylem6  9925  firest  16696  ablfaclem3  19129  ablfac2  19131  cmpcovf  21915  cmpsub  21924  tgcmp  21925  hauscmplem  21930  comppfsc  22056  ptcnplem  22145  alexsubALTlem3  22573  alexsubALT  22575  tsmsxplem1  22676  ovolicc2lem5  24037  ovolicc2  24038  limciun  24407  cvmliftlem15  32429  matunitlindflem2  34756  ptrecube  34759  istotbnd3  34917  sstotbnd2  34920  sstotbnd  34921  prdsbnd  34939  prdstotbnd  34940  heiborlem1  34957  heibor  34967  kelac1  39528  hbt  39595
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