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Theorem choicefi 41339
Description: For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
choicefi.a (𝜑𝐴 ∈ Fin)
choicefi.b ((𝜑𝑥𝐴) → 𝐵𝑊)
choicefi.n ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
Assertion
Ref Expression
choicefi (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
Distinct variable groups:   𝐴,𝑓,𝑥   𝐵,𝑓   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥)   𝑊(𝑥,𝑓)

Proof of Theorem choicefi
Dummy variables 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 choicefi.a . . . . 5 (𝜑𝐴 ∈ Fin)
2 mptfi 8811 . . . . 5 (𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
31, 2syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ Fin)
4 rnfi 8795 . . . 4 ((𝑥𝐴𝐵) ∈ Fin → ran (𝑥𝐴𝐵) ∈ Fin)
53, 4syl 17 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ∈ Fin)
6 fnchoice 41163 . . 3 (ran (𝑥𝐴𝐵) ∈ Fin → ∃𝑔(𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)))
75, 6syl 17 . 2 (𝜑 → ∃𝑔(𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)))
8 simpl 483 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → 𝜑)
9 simprl 767 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥𝐴𝐵))
10 nfv 1906 . . . . . . . 8 𝑦𝜑
11 nfra1 3216 . . . . . . . 8 𝑦𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
1210, 11nfan 1891 . . . . . . 7 𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
13 rspa 3203 . . . . . . . . . . . 12 ((∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
1413adantll 710 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
15 vex 3495 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
16 eqid 2818 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1716elrnmpt 5821 . . . . . . . . . . . . . . . 16 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
1815, 17ax-mp 5 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1918biimpi 217 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑦 = 𝐵)
2019adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑦 = 𝐵)
21 simp3 1130 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
22 choicefi.n . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
23223adant3 1124 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
2421, 23eqnetrd 3080 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 ≠ ∅)
25243exp 1111 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ≠ ∅)))
2625rexlimdv 3280 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2726adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2820, 27mpd 15 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ≠ ∅)
2928adantlr 711 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ≠ ∅)
30 id 22 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
3130imp 407 . . . . . . . . . . 11 (((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔𝑦) ∈ 𝑦)
3214, 29, 31syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑔𝑦) ∈ 𝑦)
3332ex 413 . . . . . . . . 9 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥𝐴𝐵) → (𝑔𝑦) ∈ 𝑦))
3412, 33ralrimi 3213 . . . . . . . 8 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
35 rsp 3202 . . . . . . . 8 (∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦 → (𝑦 ∈ ran (𝑥𝐴𝐵) → (𝑔𝑦) ∈ 𝑦))
3634, 35syl 17 . . . . . . 7 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥𝐴𝐵) → (𝑔𝑦) ∈ 𝑦))
3712, 36ralrimi 3213 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
3837adantrl 712 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
39 vex 3495 . . . . . . . . 9 𝑔 ∈ V
4039a1i 11 . . . . . . . 8 (𝜑𝑔 ∈ V)
411mptexd 6978 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ V)
42 coexg 7623 . . . . . . . 8 ((𝑔 ∈ V ∧ (𝑥𝐴𝐵) ∈ V) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
4340, 41, 42syl2anc 584 . . . . . . 7 (𝜑 → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
44433ad2ant1 1125 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
45 simpr 485 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → 𝑔 Fn ran (𝑥𝐴𝐵))
46 choicefi.b . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑊)
4746ralrimiva 3179 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
4816fnmpt 6481 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
4947, 48syl 17 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
5049adantr 481 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑥𝐴𝐵) Fn 𝐴)
51 ssidd 3987 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵))
52 fnco 6458 . . . . . . . . 9 ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
5345, 50, 51, 52syl3anc 1363 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
54533adant3 1124 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
55 nfv 1906 . . . . . . . . 9 𝑥𝜑
56 nfcv 2974 . . . . . . . . . 10 𝑥𝑔
57 nfmpt1 5155 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
5857nfrn 5817 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
5956, 58nffn 6445 . . . . . . . . 9 𝑥 𝑔 Fn ran (𝑥𝐴𝐵)
60 nfv 1906 . . . . . . . . . 10 𝑥(𝑔𝑦) ∈ 𝑦
6158, 60nfralw 3222 . . . . . . . . 9 𝑥𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦
6255, 59, 61nf3an 1893 . . . . . . . 8 𝑥(𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
63 funmpt 6386 . . . . . . . . . . . . . 14 Fun (𝑥𝐴𝐵)
6463a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun (𝑥𝐴𝐵))
65 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑥𝐴)
6616, 46dmmptd 6486 . . . . . . . . . . . . . . . 16 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
6766eqcomd 2824 . . . . . . . . . . . . . . 15 (𝜑𝐴 = dom (𝑥𝐴𝐵))
6867adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = dom (𝑥𝐴𝐵))
6965, 68eleqtrd 2912 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑥 ∈ dom (𝑥𝐴𝐵))
70 fvco 6752 . . . . . . . . . . . . 13 ((Fun (𝑥𝐴𝐵) ∧ 𝑥 ∈ dom (𝑥𝐴𝐵)) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7164, 69, 70syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7216fvmpt2 6771 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7365, 46, 72syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7473fveq2d 6667 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑔‘((𝑥𝐴𝐵)‘𝑥)) = (𝑔𝐵))
7571, 74eqtrd 2853 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
76753ad2antl1 1177 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
7716elrnmpt1 5823 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵𝑊) → 𝐵 ∈ ran (𝑥𝐴𝐵))
7865, 46, 77syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
79783ad2antl1 1177 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
80 simpl3 1185 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
81 fveq2 6663 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (𝑔𝑦) = (𝑔𝐵))
82 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝐵𝑦 = 𝐵)
8381, 82eleq12d 2904 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔𝐵) ∈ 𝐵))
8483rspcva 3618 . . . . . . . . . . 11 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔𝐵) ∈ 𝐵)
8579, 80, 84syl2anc 584 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → (𝑔𝐵) ∈ 𝐵)
8676, 85eqeltrd 2910 . . . . . . . . 9 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
8786ex 413 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑥𝐴 → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
8862, 87ralrimi 3213 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
8954, 88jca 512 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
90 fneq1 6437 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴))
91 nfcv 2974 . . . . . . . . . 10 𝑥𝑓
9256, 57nfco 5729 . . . . . . . . . 10 𝑥(𝑔 ∘ (𝑥𝐴𝐵))
9391, 92nfeq 2988 . . . . . . . . 9 𝑥 𝑓 = (𝑔 ∘ (𝑥𝐴𝐵))
94 fveq1 6662 . . . . . . . . . 10 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓𝑥) = ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥))
9594eleq1d 2894 . . . . . . . . 9 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9693, 95ralbid 3228 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9790, 96anbi12d 630 . . . . . . 7 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)))
9897spcegv 3594 . . . . . 6 ((𝑔 ∘ (𝑥𝐴𝐵)) ∈ V → (((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
9944, 89, 98sylc 65 . . . . 5 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1008, 9, 38, 99syl3anc 1363 . . . 4 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
101100ex 413 . . 3 (𝜑 → ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
102101exlimdv 1925 . 2 (𝜑 → (∃𝑔(𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
1037, 102mpd 15 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288  cmpt 5137  dom cdm 5548  ran crn 5549  ccom 5552  Fun wfun 6342   Fn wfn 6343  cfv 6348  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-fin 8501
This theorem is referenced by:  axccdom  41363  axccd2  41372  qndenserrnbllem  42456  hoiqssbllem3  42783
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