Theorem indexfi 8816

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 35172. (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 3740	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
3		sbceq1a 3731	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
4	1, 2, 3	cbvrexw 3388	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
5	4	ralbii 3133	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑)
6		dfsbcq 3722	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑥) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
7	6	ac6sfi 8746	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 [𝑧 / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
8	5, 7	sylan2b 596	. . 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 6487	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
11	10	ad2antrl 727	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → 𝑓 Fn 𝐴)
12		dffn4 6571	. . . . . 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 8794	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑓:𝐴–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
15	9, 13, 14	syl2anc 587	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ∈ Fin)
16		frn 6493	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
17	16	ad2antrl 727	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ran 𝑓 ⊆ 𝐵)
18		fnfvelrn 6825	. . . . . . . . 9 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
19	10, 18	sylan 583	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ran 𝑓)
20		rspesbca 3810	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ [(𝑓‘𝑥) / 𝑦]𝜑) → ∃𝑦 ∈ ran 𝑓𝜑)
21	20	ex 416	. . . . . . . 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 3144	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → (∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
24	23	imp 410	. . . . 5 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
25	24	adantl 485	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑)
26		simpr 488	. . . . . . . 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 3740	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑
30		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑓‘𝑥) = (𝑓‘𝑤))
31	30	sbceq1d 3725	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
32		sbceq1a 3731	. . . . . . . . . . . 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 3387	. . . . . . . . . 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 3173	. . . . . . . 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 3810	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥][(𝑓‘𝑤) / 𝑦]𝜑) → ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑)
38	26, 36, 37	syl2anc 587	. . . . . . 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 3149	. . . . . 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 3722	. . . . . . . . 9 ⊢ (𝑧 = (𝑓‘𝑤) → ([𝑧 / 𝑦]𝜑 ↔ [(𝑓‘𝑤) / 𝑦]𝜑))
41	40	rexbidv 3256	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑤) → (∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [(𝑓‘𝑤) / 𝑦]𝜑))
42	41	ralrn 6831	. . . . . . 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 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 1915	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑
46		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
47	46, 2	nfrex 3268	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑
48	3	rexbidv 3256	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑))
49	45, 47, 48	cbvralw 3387	. . . . 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 3940	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (𝑐 ⊆ 𝐵 ↔ ran 𝑓 ⊆ 𝐵))
52		rexeq 3359	. . . . . . 7 ⊢ (𝑐 = ran 𝑓 → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ ran 𝑓𝜑))
53	52	ralbidv 3162	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran 𝑓𝜑))
54		raleq 3358	. . . . . 6 ⊢ (𝑐 = ran 𝑓 → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ ran 𝑓∃𝑥 ∈ 𝐴 𝜑))
55	51, 53, 54	3anbi123d 1433	. . . . 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 3571	. . . 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 1369	. . 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 1128	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 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  [wsbc 3720   ⊆ wss 3881  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  Fincfn 8492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  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 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-fin 8496

This theorem is referenced by:  filbcmb  35178