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Theorem ac6sfi 8411
Description: A version of ac6s 9559 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 3286 . . . 4 (𝑢 = ∅ → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑))
2 feq2 6205 . . . . . 6 (𝑢 = ∅ → (𝑓:𝑢𝐵𝑓:∅⟶𝐵))
3 raleq 3286 . . . . . 6 (𝑢 = ∅ → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ ∅ 𝜓))
42, 3anbi12d 624 . . . . 5 (𝑢 = ∅ → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
54exbidv 2016 . . . 4 (𝑢 = ∅ → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
61, 5imbi12d 335 . . 3 (𝑢 = ∅ → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥 ∈ ∅ ∃𝑦𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))))
7 raleq 3286 . . . 4 (𝑢 = 𝑤 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝑤𝑦𝐵 𝜑))
8 feq2 6205 . . . . . 6 (𝑢 = 𝑤 → (𝑓:𝑢𝐵𝑓:𝑤𝐵))
9 raleq 3286 . . . . . 6 (𝑢 = 𝑤 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝑤 𝜓))
108, 9anbi12d 624 . . . . 5 (𝑢 = 𝑤 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
1110exbidv 2016 . . . 4 (𝑢 = 𝑤 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)))
127, 11imbi12d 335 . . 3 (𝑢 = 𝑤 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝑤𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓))))
13 raleq 3286 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑))
14 feq2 6205 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (𝑓:𝑢𝐵𝑓:(𝑤 ∪ {𝑧})⟶𝐵))
15 raleq 3286 . . . . . . 7 (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥𝑢 𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓))
1614, 15anbi12d 624 . . . . . 6 (𝑢 = (𝑤 ∪ {𝑧}) → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
1716exbidv 2016 . . . . 5 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18 feq1 6204 . . . . . . 7 (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19 fvex 6388 . . . . . . . . . 10 (𝑓𝑥) ∈ V
20 ac6sfi.1 . . . . . . . . . 10 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
2119, 20sbcie 3631 . . . . . . . . 9 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
22 fveq1 6374 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
2322sbceq1d 3601 . . . . . . . . 9 (𝑓 = 𝑔 → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔𝑥) / 𝑦]𝜑))
2421, 23syl5bbr 276 . . . . . . . 8 (𝑓 = 𝑔 → (𝜓[(𝑔𝑥) / 𝑦]𝜑))
2524ralbidv 3133 . . . . . . 7 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2618, 25anbi12d 624 . . . . . 6 (𝑓 = 𝑔 → ((𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2726cbvexvw 2137 . . . . 5 (∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
2817, 27syl6bb 278 . . . 4 (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
2913, 28imbi12d 335 . . 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 3286 . . . 4 (𝑢 = 𝐴 → (∀𝑥𝑢𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜑))
31 feq2 6205 . . . . . 6 (𝑢 = 𝐴 → (𝑓:𝑢𝐵𝑓:𝐴𝐵))
32 raleq 3286 . . . . . 6 (𝑢 = 𝐴 → (∀𝑥𝑢 𝜓 ↔ ∀𝑥𝐴 𝜓))
3331, 32anbi12d 624 . . . . 5 (𝑢 = 𝐴 → ((𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3433exbidv 2016 . . . 4 (𝑢 = 𝐴 → (∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓) ↔ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
3530, 34imbi12d 335 . . 3 (𝑢 = 𝐴 → ((∀𝑥𝑢𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝑢𝐵 ∧ ∀𝑥𝑢 𝜓)) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))))
36 f0 6268 . . . 4 ∅:∅⟶𝐵
37 0ex 4950 . . . . 5 ∅ ∈ V
38 ral0 4235 . . . . . . 7 𝑥 ∈ ∅ 𝜓
3938biantru 525 . . . . . 6 (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40 feq1 6204 . . . . . 6 (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
4139, 40syl5bbr 276 . . . . 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 3938 . . . . . . 7 𝑤 ⊆ (𝑤 ∪ {𝑧})
45 ssralv 3826 . . . . . . 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 3939 . . . . . . . . 9 {𝑧} ⊆ (𝑤 ∪ {𝑧})
49 ssralv 3826 . . . . . . . . 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 3353 . . . . . . . . . 10 𝑧 ∈ V
52 ralsnsg 4373 . . . . . . . . . 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 3672 . . . . . . . . 9 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5553, 54bitri 266 . . . . . . . 8 (∀𝑥 ∈ {𝑧}∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
5650, 55sylib 209 . . . . . . 7 (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦𝐵 𝜑 → ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
57 nfv 2009 . . . . . . . 8 𝑦 ¬ 𝑧𝑤
58 nfv 2009 . . . . . . . . 9 𝑦𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)
59 nfv 2009 . . . . . . . . . . 11 𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
60 nfcv 2907 . . . . . . . . . . . 12 𝑦(𝑤 ∪ {𝑧})
61 nfsbc1v 3616 . . . . . . . . . . . 12 𝑦[(𝑔𝑥) / 𝑦]𝜑
6260, 61nfral 3092 . . . . . . . . . . 11 𝑦𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑
6359, 62nfan 1998 . . . . . . . . . 10 𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6463nfex 2327 . . . . . . . . 9 𝑦𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)
6558, 64nfim 1995 . . . . . . . 8 𝑦(∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
66 simprl 787 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑓:𝑤𝐵)
67 vex 3353 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
6851, 67f1osn 6359 . . . . . . . . . . . . . . 15 {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69 f1of 6320 . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦𝐵)
7271snssd 4494 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
7370, 72fssd 6237 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵)
74 simpl1 1242 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ¬ 𝑧𝑤)
75 disjsn 4402 . . . . . . . . . . . . . 14 ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑤)
7674, 75sylibr 225 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑤 ∩ {𝑧}) = ∅)
77 fun2 6249 . . . . . . . . . . . . 13 (((𝑓:𝑤𝐵 ∧ {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵) ∧ (𝑤 ∩ {𝑧}) = ∅) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
7866, 73, 76, 77syl21anc 866 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
79 simprr 789 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 𝜓)
80 eleq1a 2839 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑤 → (𝑧 = 𝑥𝑧𝑤))
8180necon3bd 2951 . . . . . . . . . . . . . . . . . 18 (𝑥𝑤 → (¬ 𝑧𝑤𝑧𝑥))
8281impcom 396 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑤𝑥𝑤) → 𝑧𝑥)
83 fvunsn 6638 . . . . . . . . . . . . . . . . 17 (𝑧𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥))
84 dfsbcq 3598 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓𝑥) → ([((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑[(𝑓𝑥) / 𝑦]𝜑))
8584, 21syl6rbb 279 . . . . . . . . . . . . . . . . 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 3132 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
90 simpl3 1246 . . . . . . . . . . . . . 14 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → [𝑧 / 𝑥]𝜑)
91 ffun 6226 . . . . . . . . . . . . . . . . 17 ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
92 ssun2 3939 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
93 vsnid 4367 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
9467dmsnop 5793 . . . . . . . . . . . . . . . . . . 19 dom {⟨𝑧, 𝑦⟩} = {𝑧}
9593, 94eleqtrri 2843 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
96 funssfv 6396 . . . . . . . . . . . . . . . . . 18 ((Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
9792, 95, 96mp3an23 1577 . . . . . . . . . . . . . . . . 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 6639 . . . . . . . . . . . . . . . 16 ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
10098, 99syl6req 2816 . . . . . . . . . . . . . . 15 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
101 ralsnsg 4373 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑))
10251, 101ax-mp 5 . . . . . . . . . . . . . . . 16 (∀𝑥 ∈ {𝑧}𝜑[𝑧 / 𝑥]𝜑)
103 elsni 4351 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
104103fveq2d 6379 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
105104eqeq2d 2775 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑧} → (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) ↔ 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧)))
106105biimparc 471 . . . . . . . . . . . . . . . . . 18 ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
107 sbceq1a 3607 . . . . . . . . . . . . . . . . . 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 3132 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → (∀𝑥 ∈ {𝑧}𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
110102, 109syl5bbr 276 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 13 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
113 ralun 3957 . . . . . . . . . . . . 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 579 . . . . . . . . . . . 12 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
115 vex 3353 . . . . . . . . . . . . . 14 𝑓 ∈ V
116 snex 5064 . . . . . . . . . . . . . 14 {⟨𝑧, 𝑦⟩} ∈ V
117115, 116unex 7154 . . . . . . . . . . . . 13 (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
118 feq1 6204 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
119 fveq1 6374 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
120119sbceq1d 3601 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔𝑥) / 𝑦]𝜑[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
121120ralbidv 3133 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
122118, 121anbi12d 624 . . . . . . . . . . . . 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 579 . . . . . . . . . . 11 (((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓)) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑))
125124ex 401 . . . . . . . . . 10 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → ((𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
126125exlimdv 2028 . . . . . . . . 9 ((¬ 𝑧𝑤𝑦𝐵[𝑧 / 𝑥]𝜑) → (∃𝑓(𝑓:𝑤𝐵 ∧ ∀𝑥𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔𝑥) / 𝑦]𝜑)))
1271263exp 1148 . . . . . . . 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 3173 . . . . . . 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 473 . . 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 8408 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
134133imp 395 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  [wsbc 3596  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340  dom cdm 5277  Fun wfun 6062  wf 6064  1-1-ontowf1o 6067  cfv 6068  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-fin 8164
This theorem is referenced by:  fissuni  8478  fipreima  8479  indexfi  8481  finacn  9124  axcc4dom  9516  ttukeylem6  9589  firest  16359  ablfaclem3  18753  ablfac2  18755  cmpcovf  21474  cmpsub  21483  tgcmp  21484  hauscmplem  21489  comppfsc  21615  ptcnplem  21704  alexsubALTlem3  22132  alexsubALT  22134  tsmsxplem1  22235  ovolicc2lem5  23579  ovolicc2  23580  limciun  23949  cvmliftlem15  31728  matunitlindflem2  33830  ptrecube  33833  istotbnd3  33992  sstotbnd2  33995  sstotbnd  33996  prdsbnd  34014  prdstotbnd  34015  heiborlem1  34032  heibor  34042  kelac1  38310  hbt  38377
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