Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  choicefi Structured version   Visualization version   GIF version

Theorem choicefi 44197
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 9353 . . . . 5 (𝐴 ∈ Fin → (𝑥𝐴𝐵) ∈ Fin)
31, 2syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ Fin)
4 rnfi 9337 . . . 4 ((𝑥𝐴𝐵) ∈ Fin → ran (𝑥𝐴𝐵) ∈ Fin)
53, 4syl 17 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ∈ Fin)
6 fnchoice 44015 . . 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 481 . . . . 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 1915 . . . . . . . 8 𝑦𝜑
11 nfra1 3279 . . . . . . . 8 𝑦𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
1210, 11nfan 1900 . . . . . . 7 𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))
13 rspa 3243 . . . . . . . . . . . 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 3476 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
16 eqid 2730 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1716elrnmpt 5954 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑦 = 𝐵)
21 simp3 1136 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
22 choicefi.n . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
23223adant3 1130 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
2421, 23eqnetrd 3006 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑦 = 𝐵) → 𝑦 ≠ ∅)
25243exp 1117 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ≠ ∅)))
2625rexlimdv 3151 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ≠ ∅))
2726adantr 479 . . . . . . . . . . . . 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 405 . . . . . . . . . . 11 (((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔𝑦) ∈ 𝑦)
3214, 29, 31syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥𝐴𝐵)) → (𝑔𝑦) ∈ 𝑦)
3332ex 411 . . . . . . . . 9 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥𝐴𝐵) → (𝑔𝑦) ∈ 𝑦))
3412, 33ralrimi 3252 . . . . . . . 8 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
35 rsp 3242 . . . . . . . 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 3252 . . . . . 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 3476 . . . . . . . . 9 𝑔 ∈ V
4039a1i 11 . . . . . . . 8 (𝜑𝑔 ∈ V)
411mptexd 7227 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ V)
42 coexg 7922 . . . . . . . 8 ((𝑔 ∈ V ∧ (𝑥𝐴𝐵) ∈ V) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
4340, 41, 42syl2anc 582 . . . . . . 7 (𝜑 → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
44433ad2ant1 1131 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) ∈ V)
45 simpr 483 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → 𝑔 Fn ran (𝑥𝐴𝐵))
46 choicefi.b . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑊)
4746ralrimiva 3144 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
4816fnmpt 6689 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵𝑊 → (𝑥𝐴𝐵) Fn 𝐴)
4947, 48syl 17 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
5049adantr 479 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑥𝐴𝐵) Fn 𝐴)
51 ssidd 4004 . . . . . . . . 9 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵))
52 fnco 6666 . . . . . . . . 9 ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ ran (𝑥𝐴𝐵) ⊆ ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
5345, 50, 51, 52syl3anc 1369 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵)) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
54533adant3 1130 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴)
55 nfv 1915 . . . . . . . . 9 𝑥𝜑
56 nfcv 2901 . . . . . . . . . 10 𝑥𝑔
57 nfmpt1 5255 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
5857nfrn 5950 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
5956, 58nffn 6647 . . . . . . . . 9 𝑥 𝑔 Fn ran (𝑥𝐴𝐵)
60 nfv 1915 . . . . . . . . . 10 𝑥(𝑔𝑦) ∈ 𝑦
6158, 60nfralw 3306 . . . . . . . . 9 𝑥𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦
6255, 59, 61nf3an 1902 . . . . . . . 8 𝑥(𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
63 funmpt 6585 . . . . . . . . . . . . . 14 Fun (𝑥𝐴𝐵)
6463a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun (𝑥𝐴𝐵))
65 simpr 483 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑥𝐴)
6616, 46dmmptd 6694 . . . . . . . . . . . . . . . 16 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
6766eqcomd 2736 . . . . . . . . . . . . . . 15 (𝜑𝐴 = dom (𝑥𝐴𝐵))
6867adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = dom (𝑥𝐴𝐵))
6965, 68eleqtrd 2833 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑥 ∈ dom (𝑥𝐴𝐵))
70 fvco 6988 . . . . . . . . . . . . 13 ((Fun (𝑥𝐴𝐵) ∧ 𝑥 ∈ dom (𝑥𝐴𝐵)) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7164, 69, 70syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔‘((𝑥𝐴𝐵)‘𝑥)))
7216fvmpt2 7008 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7365, 46, 72syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7473fveq2d 6894 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑔‘((𝑥𝐴𝐵)‘𝑥)) = (𝑔𝐵))
7571, 74eqtrd 2770 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
76753ad2antl1 1183 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) = (𝑔𝐵))
7716elrnmpt1 5956 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵𝑊) → 𝐵 ∈ ran (𝑥𝐴𝐵))
7865, 46, 77syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
79783ad2antl1 1183 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
80 simpl3 1191 . . . . . . . . . . 11 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦)
81 fveq2 6890 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (𝑔𝑦) = (𝑔𝐵))
82 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝐵𝑦 = 𝐵)
8381, 82eleq12d 2825 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔𝐵) ∈ 𝐵))
8483rspcva 3609 . . . . . . . . . . 11 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑔𝐵) ∈ 𝐵)
8579, 80, 84syl2anc 582 . . . . . . . . . 10 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → (𝑔𝐵) ∈ 𝐵)
8676, 85eqeltrd 2831 . . . . . . . . 9 (((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) ∧ 𝑥𝐴) → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
8786ex 411 . . . . . . . 8 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → (𝑥𝐴 → ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
8862, 87ralrimi 3252 . . . . . . 7 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)
8954, 88jca 510 . . . . . 6 ((𝜑𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑔𝑦) ∈ 𝑦) → ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
90 fneq1 6639 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴))
91 nfcv 2901 . . . . . . . . . 10 𝑥𝑓
9256, 57nfco 5864 . . . . . . . . . 10 𝑥(𝑔 ∘ (𝑥𝐴𝐵))
9391, 92nfeq 2914 . . . . . . . . 9 𝑥 𝑓 = (𝑔 ∘ (𝑥𝐴𝐵))
94 fveq1 6889 . . . . . . . . . 10 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (𝑓𝑥) = ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥))
9594eleq1d 2816 . . . . . . . . 9 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9693, 95ralbid 3268 . . . . . . . 8 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵))
9790, 96anbi12d 629 . . . . . . 7 (𝑓 = (𝑔 ∘ (𝑥𝐴𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥𝐴𝐵)) Fn 𝐴 ∧ ∀𝑥𝐴 ((𝑔 ∘ (𝑥𝐴𝐵))‘𝑥) ∈ 𝐵)))
9897spcegv 3586 . . . . . 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 1369 . . . 4 ((𝜑 ∧ (𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
101100ex 411 . . 3 (𝜑 → ((𝑔 Fn ran (𝑥𝐴𝐵) ∧ ∀𝑦 ∈ ran (𝑥𝐴𝐵)(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)))
102101exlimdv 1934 . 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 394  w3a 1085   = wceq 1539  wex 1779  wcel 2104  wne 2938  wral 3059  wrex 3068  Vcvv 3472  wss 3947  c0 4321  cmpt 5230  dom cdm 5675  ran crn 5676  ccom 5679  Fun wfun 6536   Fn wfn 6537  cfv 6542  Fincfn 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945
This theorem is referenced by:  axccdom  44219  axccd2  44227  qndenserrnbllem  45308  hoiqssbllem3  45638
  Copyright terms: Public domain W3C validator