Theorem choicefi 42262

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	⊢ (𝜑 → 𝐴 ∈ Fin)
choicefi.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
choicefi.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)

Assertion

Ref	Expression
choicefi	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 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.

Step	Hyp	Ref	Expression
1		choicefi.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		mptfi 8889	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
4		rnfi 8873	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
6		fnchoice 42094	. . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)))
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)))
8		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝜑)
9		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵))
10		nfv 1920	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
11		nfra1 3130	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)
12	10, 11	nfan 1905	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
13		rspa 3118	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
14	13	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
15		vex 3401	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
16		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	16	elrnmpt 5793	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
18	15, 17	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
19	18	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
20	19	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
21		simp3 1139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
22		choicefi.n	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
23	22	3adant3 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
24	21, 23	eqnetrd 3001	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
25	24	3exp 1120	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑦 ≠ ∅)))
26	25	rexlimdv 3192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅))
27	26	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅))
28	20, 27	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅)
29	28	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅)
30		id 22	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
31	30	imp 410	. . . . . . . . . . 11 ⊢ (((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔‘𝑦) ∈ 𝑦)
32	14, 29, 31	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔‘𝑦) ∈ 𝑦)
33	32	ex 416	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
34	12, 33	ralrimi 3127	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
35		rsp 3117	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
36	34, 35	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
37	12, 36	ralrimi 3127	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
38	37	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
39		vex 3401	. . . . . . . . 9 ⊢ 𝑔 ∈ V
40	39	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑔 ∈ V)
41	1	mptexd 6991	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
42		coexg 7653	. . . . . . . 8 ⊢ ((𝑔 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
43	40, 41, 42	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
44	43	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
45		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵))
46		choicefi.b	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
47	46	ralrimiva 3096	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
48	16	fnmpt 6471	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
50	49	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
51		ssidd 3898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
52		fnco 6447	. . . . . . . . 9 ⊢ ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)
53	45, 50, 51, 52	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)
54	53	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)
55		nfv 1920	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
56		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑔
57		nfmpt1 5125	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
58	57	nfrn 5789	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
59	56, 58	nffn 6431	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)
60		nfv 1920	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑔‘𝑦) ∈ 𝑦
61	58, 60	nfralw 3137	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦
62	55, 59, 61	nf3an 1907	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
63		funmpt 6371	. . . . . . . . . . . . . 14 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
64	63	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
65		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
66	16, 46	dmmptd 6476	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
67	66	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
68	67	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
69	65, 68	eleqtrd 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵))
70		fvco 6760	. . . . . . . . . . . . 13 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
71	64, 69, 70	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
72	16	fvmpt2 6780	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
73	65, 46, 72	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
74	73	fveq2d 6672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑔‘𝐵))
75	71, 74	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘𝐵))
76	75	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘𝐵))
77	16	elrnmpt1 5795	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
78	65, 46, 77	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
79	78	3ad2antl1 1186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
80		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
81		fveq2 6668	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑔‘𝑦) = (𝑔‘𝐵))
82		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
83	81, 82	eleq12d 2827	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝐵) ∈ 𝐵))
84	83	rspcva 3522	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔‘𝐵) ∈ 𝐵)
85	79, 80, 84	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝐵) ∈ 𝐵)
86	76, 85	eqeltrd 2833	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)
87	86	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑥 ∈ 𝐴 → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
88	62, 87	ralrimi 3127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)
89	54, 88	jca 515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
90		fneq1 6423	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴))
91		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
92	56, 57	nfco 5702	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))
93	91, 92	nfeq 2912	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))
94		fveq1 6667	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓‘𝑥) = ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥))
95	94	eleq1d 2817	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
96	93, 95	ralbid 3144	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
97	90, 96	anbi12d 634	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)))
98	97	spcegv 3499	. . . . . 6 ⊢ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V → (((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)))
99	44, 89, 98	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
100	8, 9, 38, 99	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
101	100	ex 416	. . 3 ⊢ (𝜑 → ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)))
102	101	exlimdv 1939	. 2 ⊢ (𝜑 → (∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)))
103	7, 102	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542  ∃wex 1786   ∈ wcel 2113   ≠ wne 2934  ∀wral 3053  ∃wrex 3054  Vcvv 3397   ⊆ wss 3841  ∅c0 4209   ↦ cmpt 5107  dom cdm 5519  ran crn 5520   ∘ ccom 5523  Fun wfun 6327   Fn wfn 6328  ‘cfv 6333  Fincfn 8548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-1st 7707  df-2nd 7708  df-1o 8124  df-er 8313  df-en 8549  df-dom 8550  df-fin 8552

This theorem is referenced by:  axccdom  42284  axccd2  42291  qndenserrnbllem  43361  hoiqssbllem3  43688