Theorem indexa 37727

Description: If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

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

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

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

Proof of Theorem indexa

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

Step	Hyp	Ref	Expression
1		rabexg 5292	. 2 ⊢ (𝐵 ∈ 𝑀 → {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V)
2		ssrab2 4043	. . . 4 ⊢ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵
3	2	a1i 11	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵)
4		nfv 1914	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfre1 3262	. . . . 5 ⊢ Ⅎ𝑦∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑
6		sbceq2a 3765	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → ([𝑤 / 𝑥]𝜑 ↔ 𝜑))
7	6	rspcev 3588	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑)
8	7	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑)
9	8	anim1ci 616	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
10	9	anasss 466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
11	10	ancoms 458	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
12		sbceq2a 3765	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
13	12	sbcbidv 3809	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑤 / 𝑥]𝜑))
14	13	rexbidv 3157	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
15	14	elrab 3659	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
16	11, 15	sylibr 234	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑})
17		sbceq2a 3765	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑦]𝜑 ↔ 𝜑))
18	17	rspcev 3588	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∧ 𝜑) → ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
19	16, 18	sylancom 588	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → ∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑)
20		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑣{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
21		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
22		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑤
23		nfsbc1v 3773	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
24	22, 23	nfsbcw 3775	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
25	21, 24	nfrexw 3287	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
26		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
27	25, 26	nfrabw 3443	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
28		nfsbc1v 3773	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
29		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑣𝜑
30	20, 27, 28, 29, 17	cbvrexfw 3279	. . . . . . 7 ⊢ (∃𝑣 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}[𝑣 / 𝑦]𝜑 ↔ ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
31	19, 30	sylib 218	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
32	31	exp31 419	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)))
33	4, 5, 32	rexlimd 3244	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
34	33	ralimia 3063	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
35		nfsbc1v 3773	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
36		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
37	35, 36, 6	cbvrexw 3281	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)
38	14, 37	bitrdi 287	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
39	38	elrab 3659	. . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑))
40	39	simprbi 496	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ∃𝑥 ∈ 𝐴 𝜑)
41	40	rgen 3046	. . . 4 ⊢ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑
42	41	a1i 11	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑)
43	3, 34, 42	3jca 1128	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑))
44		sseq1 3972	. . . . 5 ⊢ (𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (𝑐 ⊆ 𝐵 ↔ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵))
45		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
46		nfsbc1v 3773	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
47	45, 46	nfrexw 3287	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
48		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
49	47, 48	nfrabw 3443	. . . . . . 7 ⊢ Ⅎ𝑥{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
50	49	nfeq2 2909	. . . . . 6 ⊢ Ⅎ𝑥 𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
51		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝑐
52	51, 27	rexeqf 3330	. . . . . 6 ⊢ (𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∃𝑦 ∈ 𝑐 𝜑 ↔ ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
53	50, 52	ralbid 3250	. . . . 5 ⊢ (𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
54	51, 27	raleqf 3329	. . . . 5 ⊢ (𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → (∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑))
55	44, 53, 54	3anbi123d 1438	. . . 4 ⊢ (𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ((𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑) ↔ ({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑)))
56	55	spcegv 3563	. . 3 ⊢ ({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V → (({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))
57	56	imp 406	. 2 ⊢ (({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V ∧ ({𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑 ∧ ∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}∃𝑥 ∈ 𝐴 𝜑)) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
58	1, 43, 57	syl2an 596	1 ⊢ ((𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447  [wsbc 3753   ⊆ wss 3914

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-sep 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-in 3921  df-ss 3931  df-pw 4565

This theorem is referenced by: (None)