Theorem indexa 36404

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 5324	. 2 ⊢ (𝐵 ∈ 𝑀 → {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ∈ V)
2		ssrab2 4073	. . . 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 1917	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
5		nfre1 3281	. . . . 5 ⊢ Ⅎ𝑦∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑
6		sbceq2a 3785	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → ([𝑤 / 𝑥]𝜑 ↔ 𝜑))
7	6	rspcev 3609	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑)
8	7	ancoms 459	. . . . . . . . . . . 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 467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
11	10	ancoms 459	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
12		sbceq2a 3785	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
13	12	sbcbidv 3832	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑤 / 𝑥]𝜑))
14	13	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
15	14	elrab 3679	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑))
16	11, 15	sylibr 233	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑})
17		sbceq2a 3785	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑦]𝜑 ↔ 𝜑))
18	17	rspcev 3609	. . . . . . . 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 2902	. . . . . . . 8 ⊢ Ⅎ𝑣{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
21		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
22		nfcv 2902	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑤
23		nfsbc1v 3793	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
24	22, 23	nfsbcw 3795	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
25	21, 24	nfrexw 3309	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
26		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
27	25, 26	nfrabw 3468	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
28		nfsbc1v 3793	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
29		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑣𝜑
30	20, 27, 28, 29, 17	cbvrexfw 3301	. . . . . . 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 217	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
32	31	exp31 420	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)))
33	4, 5, 32	rexlimd 3262	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑))
34	33	ralimia 3079	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}𝜑)
35		nfsbc1v 3793	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
36		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
37	35, 36, 6	cbvrexw 3303	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥]𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)
38	14, 37	bitrdi 286	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
39	38	elrab 3679	. . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑))
40	39	simprbi 497	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑} → ∃𝑥 ∈ 𝐴 𝜑)
41	40	rgen 3062	. . . 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 4003	. . . . 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 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
46		nfsbc1v 3793	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥][𝑧 / 𝑦]𝜑
47	45, 46	nfrexw 3309	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑
48		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
49	47, 48	nfrabw 3468	. . . . . . 7 ⊢ Ⅎ𝑥{𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
50	49	nfeq2 2919	. . . . . 6 ⊢ Ⅎ𝑥 𝑐 = {𝑧 ∈ 𝐵 ∣ ∃𝑤 ∈ 𝐴 [𝑤 / 𝑥][𝑧 / 𝑦]𝜑}
51		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑦𝑐
52	51, 27	rexeqf 3349	. . . . . 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 3269	. . . . 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 3348	. . . . 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 1436	. . . 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 3584	. . 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 407	. 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 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  {crab 3431  Vcvv 3473  [wsbc 3773   ⊆ wss 3944

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-in 3951  df-ss 3961

This theorem is referenced by: (None)