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Theorem choicefi 39706
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 8306 . . . . 5 (𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
31, 2syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ Fin)
4 rnfi 8290 . . . 4 ((𝑥𝐴𝐵) ∈ Fin → ran (𝑥𝐴𝐵) ∈ Fin)
53, 4syl 17 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ∈ Fin)
6 fnchoice 39502 . . 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 472 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → 𝜑)
9 simprl 809 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥𝐴𝐵))
10 nfv 1883 . . . . . . . 8 𝑦𝜑
11 nfra1 2970 . . . . . . . 8 𝑦𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
1210, 11nfan 1868 . . . . . . 7 𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
13 rspa 2959 . . . . . . . . . . . 12 ((∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
1413adantll 750 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
15 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
16 eqid 2651 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1716elrnmpt 5404 . . . . . . . . . . . . . . . 16 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
1815, 17ax-mp 5 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵)
1918biimpi 206 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑦 = 𝐵)
2019adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑦 = 𝐵)
21 simp3 1083 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
22 choicefi.n . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
23223adant3 1101 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
2421, 23eqnetrd 2890 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 ≠ ∅)
25243exp 1283 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ≠ ∅)))
2625rexlimdv 3059 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2726adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2820, 27mpd 15 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ≠ ∅)
2928adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ≠ ∅)
30 id 22 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → (𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
3130imp 444 . . . . . . . . . . 11 (((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔𝑦) ∈ 𝑦)
3214, 29, 31syl2anc 694 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑔𝑦) ∈ 𝑦)
3332ex 449 . . . . . . . . 9 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥𝐴𝐵) → (𝑔𝑦) ∈ 𝑦))
3412, 33ralrimi 2986 . . . . . . . 8 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
35 rsp 2958 . . . . . . . 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 2986 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
3837adantrl 752 . . . . 5 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
39 vex 3234 . . . . . . . . 9 𝑔 ∈ V
4039a1i 11 . . . . . . . 8 (𝜑𝑔 ∈ V)
411mptexd 6528 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ V)
42 coexg 7159 . . . . . . . 8 ((𝑔 ∈ V ∧ (𝑥𝐴𝐵) ∈ V) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
4340, 41, 42syl2anc 694 . . . . . . 7 (𝜑 → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
44433ad2ant1 1102 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
45 simpr 476 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → 𝑔 Fn ran (𝑥𝐴𝐵))
46 choicefi.b . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑊)
4746ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
4816fnmpt 6058 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
4947, 48syl 17 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
5049adantr 480 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑥𝐴𝐵) Fn 𝐴)
51 ssid 3657 . . . . . . . . . 10 ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵)
5251a1i 11 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵))
53 fnco 6037 . . . . . . . . 9 ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
5445, 50, 52, 53syl3anc 1366 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
55543adant3 1101 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
56 nfv 1883 . . . . . . . . 9 𝑥𝜑
57 nfcv 2793 . . . . . . . . . 10 𝑥𝑔
58 nfmpt1 4780 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
5958nfrn 5400 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
6057, 59nffn 6025 . . . . . . . . 9 𝑥 𝑔 Fn ran (𝑥𝐴𝐵)
61 nfv 1883 . . . . . . . . . 10 𝑥(𝑔𝑦) ∈ 𝑦
6259, 61nfral 2974 . . . . . . . . 9 𝑥𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦
6356, 60, 62nf3an 1871 . . . . . . . 8 𝑥(𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
64 funmpt 5964 . . . . . . . . . . . . . 14 Fun (𝑥𝐴𝐵)
6564a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun (𝑥𝐴𝐵))
66 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑥𝐴)
6716, 46dmmptd 6062 . . . . . . . . . . . . . . . 16 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
6867eqcomd 2657 . . . . . . . . . . . . . . 15 (𝜑𝐴 = dom (𝑥𝐴𝐵))
6968adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = dom (𝑥𝐴𝐵))
7066, 69eleqtrd 2732 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑥 ∈ dom (𝑥𝐴𝐵))
71 fvco 6313 . . . . . . . . . . . . 13 ((Fun (𝑥𝐴𝐵) ∧ 𝑥 ∈ dom (𝑥𝐴𝐵)) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7265, 70, 71syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7316fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7466, 46, 73syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7574fveq2d 6233 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑔‘((𝑥𝐴𝐵)‘𝑥)) = (𝑔𝐵))
7672, 75eqtrd 2685 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
77763ad2antl1 1243 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
7816elrnmpt1 5406 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵𝑊) → 𝐵 ∈ ran (𝑥𝐴𝐵))
7966, 46, 78syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
80793ad2antl1 1243 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
81 simpl3 1086 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
82 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (𝑔𝑦) = (𝑔𝐵))
83 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝐵𝑦 = 𝐵)
8482, 83eleq12d 2724 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔𝐵) ∈ 𝐵))
8584rspcva 3338 . . . . . . . . . . 11 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔𝐵) ∈ 𝐵)
8680, 81, 85syl2anc 694 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → (𝑔𝐵) ∈ 𝐵)
8777, 86eqeltrd 2730 . . . . . . . . 9 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
8887ex 449 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑥𝐴 → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
8963, 88ralrimi 2986 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
9055, 89jca 553 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
91 fneq1 6017 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴))
92 nfcv 2793 . . . . . . . . . 10 𝑥𝑓
9357, 58nfco 5320 . . . . . . . . . 10 𝑥(𝑔 ∘ (𝑥𝐴𝐵))
9492, 93nfeq 2805 . . . . . . . . 9 𝑥 𝑓 = (𝑔 ∘ (𝑥𝐴𝐵))
95 fveq1 6228 . . . . . . . . . 10 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓𝑥) = ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥))
9695eleq1d 2715 . . . . . . . . 9 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9794, 96ralbid 3012 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9891, 97anbi12d 747 . . . . . . 7 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)))
9998spcegv 3325 . . . . . 6 ((𝑔 ∘ (𝑥𝐴𝐵)) ∈ V → (((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
10044, 90, 99sylc 65 . . . . 5 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1018, 9, 38, 100syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
102101ex 449 . . 3 (𝜑 → ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
103102exlimdv 1901 . 2 (𝜑 → (∃𝑔(𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
1047, 103mpd 15 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  cmpt 4762  dom cdm 5143  ran crn 5144  ccom 5147  Fun wfun 5920   Fn wfn 5921  cfv 5926  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001
This theorem is referenced by:  axccdom  39730  axccd2  39744  qndenserrnbllem  40832  hoiqssbllem3  41159
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