Theorem indexfi 9260

Description: If for every element of a finite indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a finite subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Proven without the Axiom of Choice, unlike indexdom 37935. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
indexfi	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝑐,𝑦,𝐴   𝐵,𝑐,𝑥,𝑦   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑐)

Proof of Theorem indexfi

Dummy variables 𝑓 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧𝜑
2		nfsbc1v 3760	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
3		sbceq1a 3751	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
4	1, 2, 3	cbvrexw 3279	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
5	4	ralbii 3082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
6		dfsbcq 3742	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
7	6	ac6sfi 9184	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
8	5, 7	sylan2b 594	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
9		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝐴 ∈ Fin)
10		ffn 6662	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
11	10	ad2antrl 728	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12		dffn4 6752	. . . . . 6 ⊢ (𝑓 Fn 𝐴 ↔ 𝑓:𝐴–onto→ran 𝑓)
13	11, 12	sylib 218	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝑓:𝐴–onto→ran 𝑓)
14		fofi 9213	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑓:𝐴–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
15	9, 13, 14	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16		frn 6669	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
17	16	ad2antrl 728	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ⊆ 𝐵)
18		fnfvelrn 7025	. . . . . . . . 9 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
19	10, 18	sylan 580	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
20		rspesbca 3831	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
21	20	ex 412	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ran 𝑓 → ([(𝑓‘𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
22	19, 21	syl 17	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ([(𝑓‘𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
23	22	ralimdva 3148	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
24	23	imp 406	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
25	24	adantl 481	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
26		simpr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
27		simprr 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
28		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑤[(𝑓‘𝑥) / 𝑦]𝜑
29		nfsbc1v 3760	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑
30		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑓‘𝑥) = (𝑓‘𝑤))
31	30	sbceq1d 3745	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
32		sbceq1a 3751	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑤) / 𝑦]𝜑 ↔ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑))
33	31, 32	bitrd 279	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑))
34	28, 29, 33	cbvralw 3278	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ↔ ∀𝑤 ∈ 𝐴 [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑)
35	27, 34	sylib 218	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑤 ∈ 𝐴 [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑)
36	35	r19.21bi 3228	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) ∧ 𝑤 ∈ 𝐴) → [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑)
37		rspesbca 3831	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑) → ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑)
38	26, 36, 37	syl2anc 584	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) ∧ 𝑤 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑)
39	38	ralrimiva 3128	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑤 ∈ 𝐴 ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑)
40		dfsbcq 3742	. . . . . . . . 9 ⊢ (𝑧 = (𝑓‘𝑤) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
41	40	rexbidv 3160	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑤) → (∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
42	41	ralrn 7033	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 → (∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∀𝑤 ∈ 𝐴 ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
43	11, 42	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → (∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∀𝑤 ∈ 𝐴 ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
44	39, 43	mpbird 257	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
45		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
46		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
47	46, 2	nfrexw 3284	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
48	3	rexbidv 3160	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
49	45, 47, 48	cbvralw 3278	. . . . 5 ⊢ (∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
50	44, 49	sylibr 234	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)
51		sseq1 3959	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
52		rexeq 3292	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
53	52	ralbidv 3159	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
54		raleq 3293	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
55	51, 53, 54	3anbi123d 1438	. . . . 5 ⊢ (𝑐 = ran 𝑓 → ((𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑) ↔ (ran 𝑓 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)))
56	55	rspcev 3576	. . . 4 ⊢ ((ran 𝑓 ∈ Fin ∧ (ran 𝑓 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑 ∧ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑)) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
57	15, 17, 25, 50, 56	syl13anc 1374	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
58	8, 57	exlimddv 1936	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
59	58	3adant2 1131	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐 ∈ Fin (𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  [wsbc 3740   ⊆ wss 3901  ran crn 5625   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-dom 8885  df-fin 8887

This theorem is referenced by:  filbcmb  37941