Theorem indexfi 9305

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 38245. (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 1937	. . . . . 6 ⊢ Ⅎ𝑧𝜑
2		nfsbc1v 3767	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
3		sbceq1a 3758	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
4	1, 2, 3	cbvrexw 3308	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
5	4	ralbii 3111	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
6		dfsbcq 3749	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
7	6	ac6sfi 9232	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
8	5, 7	sylan2b 605	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
9		simpll 778	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝐴 ∈ Fin)
10		ffn 6695	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
11	10	ad2antrl 740	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12		dffn4 6788	. . . . . 6 ⊢ (𝑓 Fn 𝐴 ↔ 𝑓:𝐴–onto→ran 𝑓)
13	11, 12	sylib 221	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝑓:𝐴–onto→ran 𝑓)
14		fofi 9261	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑓:𝐴–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
15	9, 13, 14	syl2anc 595	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16		frn 6703	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
17	16	ad2antrl 740	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ⊆ 𝐵)
18		fnfvelrn 7065	. . . . . . . . 9 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
19	10, 18	sylan 591	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
20		rspesbca 3837	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
21	20	ex 417	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ran 𝑓 → ([(𝑓‘𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
22	19, 21	syl 18	. . . . . . 7 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ([(𝑓‘𝑥) / 𝑦]𝜑 → ∃𝑦 ∈ ran 𝑓𝜑))
23	22	ralimdva 3177	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
24	23	imp 411	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
25	24	adantl 486	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
26		simpr 489	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴)
27		simprr 784	. . . . . . . . . 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 1937	. . . . . . . . . . 11 ⊢ Ⅎ𝑤[(𝑓‘𝑥) / 𝑦]𝜑
29		nfsbc1v 3767	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑
30		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑓‘𝑥) = (𝑓‘𝑤))
31	30	sbceq1d 3752	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
32		sbceq1a 3758	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑤) / 𝑦]𝜑 ↔ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑))
33	31, 32	bitrd 282	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑))
34	28, 29, 33	cbvralw 3307	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 ↔ ∀𝑤 ∈ 𝐴 [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑)
35	27, 34	sylib 221	. . . . . . . . 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 3257	. . . . . . . 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 3837	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑) → ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑)
38	26, 36, 37	syl2anc 595	. . . . . . 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 3157	. . . . . 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 3749	. . . . . . . . 9 ⊢ (𝑧 = (𝑓‘𝑤) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
41	40	rexbidv 3189	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑤) → (∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
42	41	ralrn 7073	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 → (∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∀𝑤 ∈ 𝐴 ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
43	11, 42	syl 18	. . . . . 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 260	. . . . 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 1937	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
46		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
47	46, 2	nfrexw 3313	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
48	3	rexbidv 3189	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
49	45, 47, 48	cbvralw 3307	. . . . 5 ⊢ (∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ ran 𝑓∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)
50	44, 49	sylibr 237	. . . 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 3964	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
52		rexeq 3319	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
53	52	ralbidv 3188	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
54		raleq 3320	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
55	51, 53, 54	3anbi123d 1460	. . . . 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 3584	. . . 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 1395	. . 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 1958	. 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 1147	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  [wsbc 3747   ⊆ wss 3907  ran crn 5653   Fn wfn 6520  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525  Fincfn 8931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-om 7851  df-1o 8441  df-en 8932  df-dom 8933  df-fin 8935

This theorem is referenced by:  filbcmb  38251