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Theorem fineqvac 32649
Description: If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5164 and ax-pow 5242, see fineqvacALT 32650. (Contributed by BTernaryTau, 21-Sep-2024.)
Assertion
Ref Expression
fineqvac (Fin = V → CHOICE)

Proof of Theorem fineqvac
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑓 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3404 . . . . 5 𝑤 ∈ V
2 eleq2w2 2735 . . . . 5 (Fin = V → (𝑤 ∈ Fin ↔ 𝑤 ∈ V))
31, 2mpbiri 261 . . . 4 (Fin = V → 𝑤 ∈ Fin)
4 sseq2 3913 . . . . . . 7 (𝑥 = ∅ → (𝑓𝑥𝑓 ⊆ ∅))
5 dmeq 5756 . . . . . . . 8 (𝑥 = ∅ → dom 𝑥 = dom ∅)
65fneq2d 6442 . . . . . . 7 (𝑥 = ∅ → (𝑓 Fn dom 𝑥𝑓 Fn dom ∅))
74, 6anbi12d 634 . . . . . 6 (𝑥 = ∅ → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)))
87exbidv 1928 . . . . 5 (𝑥 = ∅ → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)))
9 sseq2 3913 . . . . . . 7 (𝑥 = 𝑦 → (𝑓𝑥𝑓𝑦))
10 dmeq 5756 . . . . . . . 8 (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
1110fneq2d 6442 . . . . . . 7 (𝑥 = 𝑦 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑦))
129, 11anbi12d 634 . . . . . 6 (𝑥 = 𝑦 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
1312exbidv 1928 . . . . 5 (𝑥 = 𝑦 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
14 sseq2 3913 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑓𝑥𝑓 ⊆ (𝑦 ∪ {𝑧})))
15 dmeq 5756 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → dom 𝑥 = dom (𝑦 ∪ {𝑧}))
1615fneq2d 6442 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑓 Fn dom 𝑥𝑓 Fn dom (𝑦 ∪ {𝑧})))
1714, 16anbi12d 634 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
1817exbidv 1928 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
19 sseq2 3913 . . . . . . 7 (𝑥 = 𝑤 → (𝑓𝑥𝑓𝑤))
20 dmeq 5756 . . . . . . . 8 (𝑥 = 𝑤 → dom 𝑥 = dom 𝑤)
2120fneq2d 6442 . . . . . . 7 (𝑥 = 𝑤 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑤))
2219, 21anbi12d 634 . . . . . 6 (𝑥 = 𝑤 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑤𝑓 Fn dom 𝑤)))
2322exbidv 1928 . . . . 5 (𝑥 = 𝑤 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤)))
24 ssid 3909 . . . . . 6 ∅ ⊆ ∅
25 fun0 6414 . . . . . . 7 Fun ∅
26 funfn 6379 . . . . . . 7 (Fun ∅ ↔ ∅ Fn dom ∅)
2725, 26mpbi 233 . . . . . 6 ∅ Fn dom ∅
28 0ex 5185 . . . . . . 7 ∅ ∈ V
29 sseq1 3912 . . . . . . . 8 (𝑓 = ∅ → (𝑓 ⊆ ∅ ↔ ∅ ⊆ ∅))
30 fneq1 6439 . . . . . . . 8 (𝑓 = ∅ → (𝑓 Fn dom ∅ ↔ ∅ Fn dom ∅))
3129, 30anbi12d 634 . . . . . . 7 (𝑓 = ∅ → ((𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅) ↔ (∅ ⊆ ∅ ∧ ∅ Fn dom ∅)))
3228, 31spcev 3513 . . . . . 6 ((∅ ⊆ ∅ ∧ ∅ Fn dom ∅) → ∃𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅))
3324, 27, 32mp2an 692 . . . . 5 𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)
34 sseq1 3912 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓𝑦𝑔𝑦))
35 fneq1 6439 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 Fn dom 𝑦𝑔 Fn dom 𝑦))
3634, 35anbi12d 634 . . . . . . . 8 (𝑓 = 𝑔 → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑔𝑦𝑔 Fn dom 𝑦)))
3736cbvexvw 2049 . . . . . . 7 (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
38 ssun3 4074 . . . . . . . . . . 11 (𝑔𝑦𝑔 ⊆ (𝑦 ∪ {𝑧}))
3938ad2antrr 726 . . . . . . . . . 10 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} = ∅) → 𝑔 ⊆ (𝑦 ∪ {𝑧}))
40 dmun 5763 . . . . . . . . . . . . . 14 dom (𝑦 ∪ {𝑧}) = (dom 𝑦 ∪ dom {𝑧})
41 uneq2 4057 . . . . . . . . . . . . . . 15 (dom {𝑧} = ∅ → (dom 𝑦 ∪ dom {𝑧}) = (dom 𝑦 ∪ ∅))
42 un0 4289 . . . . . . . . . . . . . . 15 (dom 𝑦 ∪ ∅) = dom 𝑦
4341, 42eqtrdi 2790 . . . . . . . . . . . . . 14 (dom {𝑧} = ∅ → (dom 𝑦 ∪ dom {𝑧}) = dom 𝑦)
4440, 43syl5eq 2786 . . . . . . . . . . . . 13 (dom {𝑧} = ∅ → dom (𝑦 ∪ {𝑧}) = dom 𝑦)
4544fneq2d 6442 . . . . . . . . . . . 12 (dom {𝑧} = ∅ → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn dom 𝑦))
4645biimparc 483 . . . . . . . . . . 11 ((𝑔 Fn dom 𝑦 ∧ dom {𝑧} = ∅) → 𝑔 Fn dom (𝑦 ∪ {𝑧}))
4746adantll 714 . . . . . . . . . 10 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} = ∅) → 𝑔 Fn dom (𝑦 ∪ {𝑧}))
48 vex 3404 . . . . . . . . . . 11 𝑔 ∈ V
49 sseq1 3912 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 ⊆ (𝑦 ∪ {𝑧}) ↔ 𝑔 ⊆ (𝑦 ∪ {𝑧})))
50 fneq1 6439 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn dom (𝑦 ∪ {𝑧})))
5149, 50anbi12d 634 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})) ↔ (𝑔 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑔 Fn dom (𝑦 ∪ {𝑧}))))
5248, 51spcev 3513 . . . . . . . . . 10 ((𝑔 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑔 Fn dom (𝑦 ∪ {𝑧})) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
5339, 47, 52syl2anc 587 . . . . . . . . 9 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} = ∅) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
54 dmsnn0 6049 . . . . . . . . . . . . 13 (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
55 elvv 5607 . . . . . . . . . . . . 13 (𝑧 ∈ (V × V) ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
5654, 55bitr3i 280 . . . . . . . . . . . 12 (dom {𝑧} ≠ ∅ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
5756anbi2i 626 . . . . . . . . . . 11 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) ↔ ((𝑔𝑦𝑔 Fn dom 𝑦) ∧ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
58 19.42vv 1965 . . . . . . . . . . 11 (∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) ↔ ((𝑔𝑦𝑔 Fn dom 𝑦) ∧ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
5957, 58bitr4i 281 . . . . . . . . . 10 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) ↔ ∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩))
60383ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → 𝑔 ⊆ (𝑦 ∪ {𝑧}))
61 snssi 4706 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ dom 𝑦 → {𝑢} ⊆ dom 𝑦)
62 ssequn2 4083 . . . . . . . . . . . . . . . . . . . . 21 ({𝑢} ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ {𝑢}) = dom 𝑦)
6361, 62sylib 221 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ dom 𝑦 → (dom 𝑦 ∪ {𝑢}) = dom 𝑦)
6463fneq2d 6442 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ dom 𝑦 → (𝑔 Fn (dom 𝑦 ∪ {𝑢}) ↔ 𝑔 Fn dom 𝑦))
6564biimparc 483 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦𝑢 ∈ dom 𝑦) → 𝑔 Fn (dom 𝑦 ∪ {𝑢}))
66653adant2 1132 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → 𝑔 Fn (dom 𝑦 ∪ {𝑢}))
67 sneq 4536 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
6867dmeqd 5758 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = ⟨𝑢, 𝑣⟩ → dom {𝑧} = dom {⟨𝑢, 𝑣⟩})
69 vex 3404 . . . . . . . . . . . . . . . . . . . . . . 23 𝑣 ∈ V
7069dmsnop 6058 . . . . . . . . . . . . . . . . . . . . . 22 dom {⟨𝑢, 𝑣⟩} = {𝑢}
7168, 70eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ⟨𝑢, 𝑣⟩ → dom {𝑧} = {𝑢})
7271uneq2d 4063 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ⟨𝑢, 𝑣⟩ → (dom 𝑦 ∪ dom {𝑧}) = (dom 𝑦 ∪ {𝑢}))
7340, 72syl5eq 2786 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨𝑢, 𝑣⟩ → dom (𝑦 ∪ {𝑧}) = (dom 𝑦 ∪ {𝑢}))
7473fneq2d 6442 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn (dom 𝑦 ∪ {𝑢})))
75743ad2ant2 1135 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn (dom 𝑦 ∪ {𝑢})))
7666, 75mpbird 260 . . . . . . . . . . . . . . . 16 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → 𝑔 Fn dom (𝑦 ∪ {𝑧}))
77763expia 1122 . . . . . . . . . . . . . . 15 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑢 ∈ dom 𝑦𝑔 Fn dom (𝑦 ∪ {𝑧})))
78773adant1 1131 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑢 ∈ dom 𝑦𝑔 Fn dom (𝑦 ∪ {𝑧})))
7960, 78, 52syl6an 684 . . . . . . . . . . . . 13 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑢 ∈ dom 𝑦 → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
8067uneq2d 4063 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑔 ∪ {𝑧}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩}))
8180adantl 485 . . . . . . . . . . . . . . . 16 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {𝑧}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩}))
82 unss1 4079 . . . . . . . . . . . . . . . . 17 (𝑔𝑦 → (𝑔 ∪ {𝑧}) ⊆ (𝑦 ∪ {𝑧}))
8382adantr 484 . . . . . . . . . . . . . . . 16 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {𝑧}) ⊆ (𝑦 ∪ {𝑧}))
8481, 83eqsstrrd 3926 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}))
85843adant2 1132 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}))
86 vex 3404 . . . . . . . . . . . . . . . . . . 19 𝑢 ∈ V
8786a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → 𝑢 ∈ V)
8869a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → 𝑣 ∈ V)
89 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → 𝑔 Fn dom 𝑦)
90 eqid 2739 . . . . . . . . . . . . . . . . . 18 (𝑔 ∪ {⟨𝑢, 𝑣⟩}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩})
91 eqid 2739 . . . . . . . . . . . . . . . . . 18 (dom 𝑦 ∪ {𝑢}) = (dom 𝑦 ∪ {𝑢})
92 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → ¬ 𝑢 ∈ dom 𝑦)
9387, 88, 89, 90, 91, 92fnunop 6461 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢}))
9493ex 416 . . . . . . . . . . . . . . . 16 (𝑔 Fn dom 𝑦 → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
95943ad2ant2 1135 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
9673fneq2d 6442 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
97963ad2ant3 1136 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → ((𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
9895, 97sylibrd 262 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧})))
99 snex 5308 . . . . . . . . . . . . . . . 16 {⟨𝑢, 𝑣⟩} ∈ V
10048, 99unex 7499 . . . . . . . . . . . . . . 15 (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ∈ V
101 sseq1 3912 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔 ∪ {⟨𝑢, 𝑣⟩}) → (𝑓 ⊆ (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧})))
102 fneq1 6439 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔 ∪ {⟨𝑢, 𝑣⟩}) → (𝑓 Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧})))
103101, 102anbi12d 634 . . . . . . . . . . . . . . 15 (𝑓 = (𝑔 ∪ {⟨𝑢, 𝑣⟩}) → ((𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})) ↔ ((𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧}))))
104100, 103spcev 3513 . . . . . . . . . . . . . 14 (((𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧})) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
10585, 98, 104syl6an 684 . . . . . . . . . . . . 13 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (¬ 𝑢 ∈ dom 𝑦 → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
10679, 105pm2.61d 182 . . . . . . . . . . . 12 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
1071063expa 1119 . . . . . . . . . . 11 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
108107exlimivv 1939 . . . . . . . . . 10 (∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
10959, 108sylbi 220 . . . . . . . . 9 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
11053, 109pm2.61dane 3022 . . . . . . . 8 ((𝑔𝑦𝑔 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
111110exlimiv 1937 . . . . . . 7 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
11237, 111sylbi 220 . . . . . 6 (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
113112a1i 11 . . . . 5 (𝑦 ∈ Fin → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
1148, 13, 18, 23, 33, 113findcard2 8775 . . . 4 (𝑤 ∈ Fin → ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
1153, 114syl 17 . . 3 (Fin = V → ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
116115alrimiv 1934 . 2 (Fin = V → ∀𝑤𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
117 df-ac 9628 . 2 (CHOICE ↔ ∀𝑤𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
118116, 117sylibr 237 1 (Fin = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wex 1786  wcel 2114  wne 2935  Vcvv 3400  cun 3851  wss 3853  c0 4221  {csn 4526  cop 4532   × cxp 5533  dom cdm 5535  Fun wfun 6343   Fn wfn 6344  Fincfn 8567  CHOICEwac 9627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-en 8568  df-fin 8571  df-ac 9628
This theorem is referenced by: (None)
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