Theorem choicefi 45187

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	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		mptfi 9278	. . 3 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
3		rnfi 9267	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin)
4		fnchoice 45016	. . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)))
5	1, 2, 3, 4	4syl 19	. 2 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)))
6		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝜑)
7		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵))
8		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
9		nfra1 3259	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)
10	8, 9	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
11		rspa 3224	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
12	11	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
13		vex 3448	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
14		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
15	14	elrnmpt 5911	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
16	13, 15	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
17	16	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
18	17	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
19		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20		choicefi.n	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
21	20	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
22	19, 21	eqnetrd 2992	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
23	22	3exp 1119	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑦 ≠ ∅)))
24	23	rexlimdv 3132	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅))
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 → 𝑦 ≠ ∅))
26	18, 25	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅)
27	26	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≠ ∅)
28		id 22	. . . . . . . . . . . 12 ⊢ ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → (𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦))
29	28	imp 406	. . . . . . . . . . 11 ⊢ (((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ 𝑦 ≠ ∅) → (𝑔‘𝑦) ∈ 𝑦)
30	12, 27, 29	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔‘𝑦) ∈ 𝑦)
31	30	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
32	10, 31	ralrimi 3233	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
33		rsp 3223	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦 → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
34	32, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑦) ∈ 𝑦))
35	10, 34	ralrimi 3233	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
36	35	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)
37		vex 3448	. . . . . . . . 9 ⊢ 𝑔 ∈ V
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑔 ∈ V)
39	1	mptexd 7180	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
40		coexg 7885	. . . . . . . 8 ⊢ ((𝑔 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
41	38, 39, 40	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
42	41	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ V)
43		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵))
44		choicefi.b	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
45	44	ralrimiva 3125	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
46	14	fnmpt 6640	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
48	47	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
49		ssidd 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
50		fnco 6618	. . . . . . . . 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)
51	43, 48, 49, 50	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)
52	51	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴)
53		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
54		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑔
55		nfmpt1 5201	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
56	55	nfrn 5905	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
57	54, 56	nffn 6599	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵)
58		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑔‘𝑦) ∈ 𝑦
59	56, 58	nfralw 3283	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦
60	53, 57, 59	nf3an 1901	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦)
61		funmpt 6538	. . . . . . . . . . . . . 14 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun (𝑥 ∈ 𝐴 ↦ 𝐵))
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
64	14, 44	dmmptd 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
65	64	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
67	63, 66	eleqtrd 2830	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵))
68		fvco 6941	. . . . . . . . . . . . 13 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
69	62, 67, 68	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
70	14	fvmpt2 6961	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
71	63, 44, 70	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
72	71	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑔‘((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑔‘𝐵))
73	69, 72	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) = (𝑔‘𝐵))
74	73	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))
75	14	elrnmpt1 5913	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
76	63, 44, 75	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
77	76	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))
78		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)
79		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑔‘𝑦) = (𝑔‘𝐵))
80		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
81	79, 80	eleq12d 2822	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘𝐵) ∈ 𝐵))
82	81	rspcva 3583	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑔‘𝐵) ∈ 𝐵)
83	77, 78, 82	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝐵) ∈ 𝐵)
84	74, 83	eqeltrd 2828	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)
85	84	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → (𝑥 ∈ 𝐴 → ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
86	60, 85	ralrimi 3233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)(𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵)
87	52, 86	jca 511	. . . . . 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))
88		fneq1 6591	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓 Fn 𝐴 ↔ (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) Fn 𝐴))
89		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
90	54, 55	nfco 5819	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))
91	89, 90	nfeq 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))
92		fveq1 6839	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (𝑓‘𝑥) = ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥))
93	92	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐵 ↔ ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
94	91, 93	ralbid 3248	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑥) ∈ 𝐵))
95	88, 94	anbi12d 632	. . . . . . 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)))
96	95	spcegv 3560	. . . . . 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)))
97	42, 87, 96	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))
98	6, 7, 36, 97	syl3anc 1373	. . . 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))
99	98	ex 412	. . 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)))
100	99	exlimdv 1933	. 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)))
101	5, 100	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  ∅c0 4292   ↦ cmpt 5183  dom cdm 5631  ran crn 5632   ∘ ccom 5635  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  Fincfn 8895

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1st 7947  df-2nd 7948  df-1o 8411  df-en 8896  df-dom 8897  df-fin 8899

This theorem is referenced by:  axccdom  45209  axccd2  45217  qndenserrnbllem  46285  hoiqssbllem3  46615