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Theorem fineqvac 35090
Description: If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5285 and ax-pow 5371, see fineqvacALT 35091. (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 3482 . . . . 5 𝑤 ∈ V
2 eleq2w2 2731 . . . . 5 (Fin = V → (𝑤 ∈ Fin ↔ 𝑤 ∈ V))
31, 2mpbiri 258 . . . 4 (Fin = V → 𝑤 ∈ Fin)
4 sseq2 4022 . . . . . . 7 (𝑥 = ∅ → (𝑓𝑥𝑓 ⊆ ∅))
5 dmeq 5917 . . . . . . . 8 (𝑥 = ∅ → dom 𝑥 = dom ∅)
65fneq2d 6663 . . . . . . 7 (𝑥 = ∅ → (𝑓 Fn dom 𝑥𝑓 Fn dom ∅))
74, 6anbi12d 632 . . . . . 6 (𝑥 = ∅ → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)))
87exbidv 1919 . . . . 5 (𝑥 = ∅ → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)))
9 sseq2 4022 . . . . . . 7 (𝑥 = 𝑦 → (𝑓𝑥𝑓𝑦))
10 dmeq 5917 . . . . . . . 8 (𝑥 = 𝑦 → dom 𝑥 = dom 𝑦)
1110fneq2d 6663 . . . . . . 7 (𝑥 = 𝑦 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑦))
129, 11anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
1312exbidv 1919 . . . . 5 (𝑥 = 𝑦 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
14 sseq2 4022 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑓𝑥𝑓 ⊆ (𝑦 ∪ {𝑧})))
15 dmeq 5917 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → dom 𝑥 = dom (𝑦 ∪ {𝑧}))
1615fneq2d 6663 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑓 Fn dom 𝑥𝑓 Fn dom (𝑦 ∪ {𝑧})))
1714, 16anbi12d 632 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
1817exbidv 1919 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧}))))
19 sseq2 4022 . . . . . . 7 (𝑥 = 𝑤 → (𝑓𝑥𝑓𝑤))
20 dmeq 5917 . . . . . . . 8 (𝑥 = 𝑤 → dom 𝑥 = dom 𝑤)
2120fneq2d 6663 . . . . . . 7 (𝑥 = 𝑤 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑤))
2219, 21anbi12d 632 . . . . . 6 (𝑥 = 𝑤 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑤𝑓 Fn dom 𝑤)))
2322exbidv 1919 . . . . 5 (𝑥 = 𝑤 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤)))
24 ssid 4018 . . . . . 6 ∅ ⊆ ∅
25 fun0 6633 . . . . . . 7 Fun ∅
26 funfn 6598 . . . . . . 7 (Fun ∅ ↔ ∅ Fn dom ∅)
2725, 26mpbi 230 . . . . . 6 ∅ Fn dom ∅
28 0ex 5313 . . . . . . 7 ∅ ∈ V
29 sseq1 4021 . . . . . . . 8 (𝑓 = ∅ → (𝑓 ⊆ ∅ ↔ ∅ ⊆ ∅))
30 fneq1 6660 . . . . . . . 8 (𝑓 = ∅ → (𝑓 Fn dom ∅ ↔ ∅ Fn dom ∅))
3129, 30anbi12d 632 . . . . . . 7 (𝑓 = ∅ → ((𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅) ↔ (∅ ⊆ ∅ ∧ ∅ Fn dom ∅)))
3228, 31spcev 3606 . . . . . 6 ((∅ ⊆ ∅ ∧ ∅ Fn dom ∅) → ∃𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅))
3324, 27, 32mp2an 692 . . . . 5 𝑓(𝑓 ⊆ ∅ ∧ 𝑓 Fn dom ∅)
34 sseq1 4021 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓𝑦𝑔𝑦))
35 fneq1 6660 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 Fn dom 𝑦𝑔 Fn dom 𝑦))
3634, 35anbi12d 632 . . . . . . . 8 (𝑓 = 𝑔 → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑔𝑦𝑔 Fn dom 𝑦)))
3736cbvexvw 2034 . . . . . . 7 (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
38 ssun3 4190 . . . . . . . . . . 11 (𝑔𝑦𝑔 ⊆ (𝑦 ∪ {𝑧}))
3938ad2antrr 726 . . . . . . . . . 10 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} = ∅) → 𝑔 ⊆ (𝑦 ∪ {𝑧}))
40 dmun 5924 . . . . . . . . . . . . . 14 dom (𝑦 ∪ {𝑧}) = (dom 𝑦 ∪ dom {𝑧})
41 uneq2 4172 . . . . . . . . . . . . . . 15 (dom {𝑧} = ∅ → (dom 𝑦 ∪ dom {𝑧}) = (dom 𝑦 ∪ ∅))
42 un0 4400 . . . . . . . . . . . . . . 15 (dom 𝑦 ∪ ∅) = dom 𝑦
4341, 42eqtrdi 2791 . . . . . . . . . . . . . 14 (dom {𝑧} = ∅ → (dom 𝑦 ∪ dom {𝑧}) = dom 𝑦)
4440, 43eqtrid 2787 . . . . . . . . . . . . 13 (dom {𝑧} = ∅ → dom (𝑦 ∪ {𝑧}) = dom 𝑦)
4544fneq2d 6663 . . . . . . . . . . . 12 (dom {𝑧} = ∅ → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn dom 𝑦))
4645biimparc 479 . . . . . . . . . . 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 3482 . . . . . . . . . . 11 𝑔 ∈ V
49 sseq1 4021 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 ⊆ (𝑦 ∪ {𝑧}) ↔ 𝑔 ⊆ (𝑦 ∪ {𝑧})))
50 fneq1 6660 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn dom (𝑦 ∪ {𝑧})))
5149, 50anbi12d 632 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})) ↔ (𝑔 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑔 Fn dom (𝑦 ∪ {𝑧}))))
5248, 51spcev 3606 . . . . . . . . . 10 ((𝑔 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑔 Fn dom (𝑦 ∪ {𝑧})) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
5339, 47, 52syl2anc 584 . . . . . . . . 9 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} = ∅) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
54 dmsnn0 6229 . . . . . . . . . . . . 13 (𝑧 ∈ (V × V) ↔ dom {𝑧} ≠ ∅)
55 elvv 5763 . . . . . . . . . . . . 13 (𝑧 ∈ (V × V) ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
5654, 55bitr3i 277 . . . . . . . . . . . 12 (dom {𝑧} ≠ ∅ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
5756anbi2i 623 . . . . . . . . . . 11 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) ↔ ((𝑔𝑦𝑔 Fn dom 𝑦) ∧ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
58 19.42vv 1955 . . . . . . . . . . 11 (∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) ↔ ((𝑔𝑦𝑔 Fn dom 𝑦) ∧ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
5957, 58bitr4i 278 . . . . . . . . . 10 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) ↔ ∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩))
60383ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → 𝑔 ⊆ (𝑦 ∪ {𝑧}))
61 snssi 4813 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ dom 𝑦 → {𝑢} ⊆ dom 𝑦)
62 ssequn2 4199 . . . . . . . . . . . . . . . . . . . . 21 ({𝑢} ⊆ dom 𝑦 ↔ (dom 𝑦 ∪ {𝑢}) = dom 𝑦)
6361, 62sylib 218 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ dom 𝑦 → (dom 𝑦 ∪ {𝑢}) = dom 𝑦)
6463fneq2d 6663 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ dom 𝑦 → (𝑔 Fn (dom 𝑦 ∪ {𝑢}) ↔ 𝑔 Fn dom 𝑦))
6564biimparc 479 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦𝑢 ∈ dom 𝑦) → 𝑔 Fn (dom 𝑦 ∪ {𝑢}))
66653adant2 1130 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → 𝑔 Fn (dom 𝑦 ∪ {𝑢}))
67 sneq 4641 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = ⟨𝑢, 𝑣⟩ → {𝑧} = {⟨𝑢, 𝑣⟩})
6867dmeqd 5919 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = ⟨𝑢, 𝑣⟩ → dom {𝑧} = dom {⟨𝑢, 𝑣⟩})
69 vex 3482 . . . . . . . . . . . . . . . . . . . . . . 23 𝑣 ∈ V
7069dmsnop 6238 . . . . . . . . . . . . . . . . . . . . . 22 dom {⟨𝑢, 𝑣⟩} = {𝑢}
7168, 70eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ⟨𝑢, 𝑣⟩ → dom {𝑧} = {𝑢})
7271uneq2d 4178 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ⟨𝑢, 𝑣⟩ → (dom 𝑦 ∪ dom {𝑧}) = (dom 𝑦 ∪ {𝑢}))
7340, 72eqtrid 2787 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨𝑢, 𝑣⟩ → dom (𝑦 ∪ {𝑧}) = (dom 𝑦 ∪ {𝑢}))
7473fneq2d 6663 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn (dom 𝑦 ∪ {𝑢})))
75743ad2ant2 1133 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → (𝑔 Fn dom (𝑦 ∪ {𝑧}) ↔ 𝑔 Fn (dom 𝑦 ∪ {𝑢})))
7666, 75mpbird 257 . . . . . . . . . . . . . . . 16 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩ ∧ 𝑢 ∈ dom 𝑦) → 𝑔 Fn dom (𝑦 ∪ {𝑧}))
77763expia 1120 . . . . . . . . . . . . . . 15 ((𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑢 ∈ dom 𝑦𝑔 Fn dom (𝑦 ∪ {𝑧})))
78773adant1 1129 . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑔 ∪ {𝑧}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩}))
8180adantl 481 . . . . . . . . . . . . . . . 16 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {𝑧}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩}))
82 unss1 4195 . . . . . . . . . . . . . . . . 17 (𝑔𝑦 → (𝑔 ∪ {𝑧}) ⊆ (𝑦 ∪ {𝑧}))
8382adantr 480 . . . . . . . . . . . . . . . 16 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {𝑧}) ⊆ (𝑦 ∪ {𝑧}))
8481, 83eqsstrrd 4035 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}))
85843adant2 1130 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧}))
86 vex 3482 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → 𝑔 Fn dom 𝑦)
90 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑔 ∪ {⟨𝑢, 𝑣⟩}) = (𝑔 ∪ {⟨𝑢, 𝑣⟩})
91 eqid 2735 . . . . . . . . . . . . . . . . . 18 (dom 𝑦 ∪ {𝑢}) = (dom 𝑦 ∪ {𝑢})
92 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → ¬ 𝑢 ∈ dom 𝑦)
9387, 88, 89, 90, 91, 92fnunop 6685 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn dom 𝑦 ∧ ¬ 𝑢 ∈ dom 𝑦) → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢}))
9493ex 412 . . . . . . . . . . . . . . . 16 (𝑔 Fn dom 𝑦 → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
95943ad2ant2 1133 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
9673fneq2d 6663 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
97963ad2ant3 1134 . . . . . . . . . . . . . . 15 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → ((𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn (dom 𝑦 ∪ {𝑢})))
9895, 97sylibrd 259 . . . . . . . . . . . . . 14 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → (¬ 𝑢 ∈ dom 𝑦 → (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧})))
99 snex 5442 . . . . . . . . . . . . . . . 16 {⟨𝑢, 𝑣⟩} ∈ V
10048, 99unex 7763 . . . . . . . . . . . . . . 15 (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ∈ V
101 sseq1 4021 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔 ∪ {⟨𝑢, 𝑣⟩}) → (𝑓 ⊆ (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) ⊆ (𝑦 ∪ {𝑧})))
102 fneq1 6660 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔 ∪ {⟨𝑢, 𝑣⟩}) → (𝑓 Fn dom (𝑦 ∪ {𝑧}) ↔ (𝑔 ∪ {⟨𝑢, 𝑣⟩}) Fn dom (𝑦 ∪ {𝑧})))
103101, 102anbi12d 632 . . . . . . . . . . . . . . 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 3606 . . . . . . . . . . . . . 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 179 . . . . . . . . . . . 12 ((𝑔𝑦𝑔 Fn dom 𝑦𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
1071063expa 1117 . . . . . . . . . . 11 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
108107exlimivv 1930 . . . . . . . . . 10 (∃𝑢𝑣((𝑔𝑦𝑔 Fn dom 𝑦) ∧ 𝑧 = ⟨𝑢, 𝑣⟩) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
10959, 108sylbi 217 . . . . . . . . 9 (((𝑔𝑦𝑔 Fn dom 𝑦) ∧ dom {𝑧} ≠ ∅) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
11053, 109pm2.61dane 3027 . . . . . . . 8 ((𝑔𝑦𝑔 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
111110exlimiv 1928 . . . . . . 7 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ (𝑦 ∪ {𝑧}) ∧ 𝑓 Fn dom (𝑦 ∪ {𝑧})))
11237, 111sylbi 217 . . . . . 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 9203 . . . 4 (𝑤 ∈ Fin → ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
1153, 114syl 17 . . 3 (Fin = V → ∃𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
116115alrimiv 1925 . 2 (Fin = V → ∀𝑤𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
117 df-ac 10154 . 2 (CHOICE ↔ ∀𝑤𝑓(𝑓𝑤𝑓 Fn dom 𝑤))
118116, 117sylibr 234 1 (Fin = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637   × cxp 5687  dom cdm 5689  Fun wfun 6557   Fn wfn 6558  Fincfn 8984  CHOICEwac 10153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-en 8985  df-fin 8988  df-ac 10154
This theorem is referenced by: (None)
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