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Theorem choicefi 43411
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 9295 . . . . 5 (𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
31, 2syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ Fin)
4 rnfi 9279 . . . 4 ((𝑥𝐴𝐵) ∈ Fin → ran (𝑥𝐴𝐵) ∈ Fin)
53, 4syl 17 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ∈ Fin)
6 fnchoice 43224 . . 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 769 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥𝐴𝐵))
10 nfv 1917 . . . . . . . 8 𝑦𝜑
11 nfra1 3267 . . . . . . . 8 𝑦𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
1210, 11nfan 1902 . . . . . . 7 𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
13 rspa 3231 . . . . . . . . . . . 12 ((∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
1413adantll 712 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
15 vex 3449 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
16 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1716elrnmpt 5911 . . . . . . . . . . . . . . . 16 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
1815, 17ax-mp 5 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1918biimpi 215 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑦 = 𝐵)
2019adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑦 = 𝐵)
21 simp3 1138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
22 choicefi.n . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
23223adant3 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
2421, 23eqnetrd 3011 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 ≠ ∅)
25243exp 1119 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ≠ ∅)))
2625rexlimdv 3150 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2726adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2820, 27mpd 15 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ≠ ∅)
2928adantlr 713 . . . . . . . . . . 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 3240 . . . . . . . 8 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
35 rsp 3230 . . . . . . . 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 3240 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
3837adantrl 714 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
39 vex 3449 . . . . . . . . 9 𝑔 ∈ V
4039a1i 11 . . . . . . . 8 (𝜑𝑔 ∈ V)
411mptexd 7174 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ V)
42 coexg 7866 . . . . . . . 8 ((𝑔 ∈ V ∧ (𝑥𝐴𝐵) ∈ V) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
4340, 41, 42syl2anc 584 . . . . . . 7 (𝜑 → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
44433ad2ant1 1133 . . . . . 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 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
4816fnmpt 6641 . . . . . . . . . . 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 3967 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵))
52 fnco 6618 . . . . . . . . 9 ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
5345, 50, 51, 52syl3anc 1371 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
54533adant3 1132 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
55 nfv 1917 . . . . . . . . 9 𝑥𝜑
56 nfcv 2907 . . . . . . . . . 10 𝑥𝑔
57 nfmpt1 5213 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
5857nfrn 5907 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
5956, 58nffn 6601 . . . . . . . . 9 𝑥 𝑔 Fn ran (𝑥𝐴𝐵)
60 nfv 1917 . . . . . . . . . 10 𝑥(𝑔𝑦) ∈ 𝑦
6158, 60nfralw 3294 . . . . . . . . 9 𝑥𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦
6255, 59, 61nf3an 1904 . . . . . . . 8 𝑥(𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
63 funmpt 6539 . . . . . . . . . . . . . 14 Fun (𝑥𝐴𝐵)
6463a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun (𝑥𝐴𝐵))
65 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑥𝐴)
6616, 46dmmptd 6646 . . . . . . . . . . . . . . . 16 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
6766eqcomd 2742 . . . . . . . . . . . . . . 15 (𝜑𝐴 = dom (𝑥𝐴𝐵))
6867adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = dom (𝑥𝐴𝐵))
6965, 68eleqtrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑥 ∈ dom (𝑥𝐴𝐵))
70 fvco 6939 . . . . . . . . . . . . 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 6959 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7365, 46, 72syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7473fveq2d 6846 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑔‘((𝑥𝐴𝐵)‘𝑥)) = (𝑔𝐵))
7571, 74eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
76753ad2antl1 1185 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
7716elrnmpt1 5913 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵𝑊) → 𝐵 ∈ ran (𝑥𝐴𝐵))
7865, 46, 77syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
79783ad2antl1 1185 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
80 simpl3 1193 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
81 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (𝑔𝑦) = (𝑔𝐵))
82 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝐵𝑦 = 𝐵)
8381, 82eleq12d 2832 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔𝐵) ∈ 𝐵))
8483rspcva 3579 . . . . . . . . . . 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 2838 . . . . . . . . 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 3240 . . . . . . 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 6593 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴))
91 nfcv 2907 . . . . . . . . . 10 𝑥𝑓
9256, 57nfco 5821 . . . . . . . . . 10 𝑥(𝑔 ∘ (𝑥𝐴𝐵))
9391, 92nfeq 2920 . . . . . . . . 9 𝑥 𝑓 = (𝑔 ∘ (𝑥𝐴𝐵))
94 fveq1 6841 . . . . . . . . . 10 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓𝑥) = ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥))
9594eleq1d 2822 . . . . . . . . 9 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9693, 95ralbid 3256 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9790, 96anbi12d 631 . . . . . . 7 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)))
9897spcegv 3556 . . . . . 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 1371 . . . 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 1936 . 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  cmpt 5188  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490   Fn wfn 6491  cfv 6496  Fincfn 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887
This theorem is referenced by:  axccdom  43433  axccd2  43441  qndenserrnbllem  44525  hoiqssbllem3  44855
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