Theorem cbvmpox 7255

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7256 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
cbvmpox.1	⊢ Ⅎ𝑧𝐵
cbvmpox.2	⊢ Ⅎ𝑥𝐷
cbvmpox.3	⊢ Ⅎ𝑧𝐶
cbvmpox.4	⊢ Ⅎ𝑤𝐶
cbvmpox.5	⊢ Ⅎ𝑥𝐸
cbvmpox.6	⊢ Ⅎ𝑦𝐸
cbvmpox.7	⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)
cbvmpox.8	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpox	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵   𝑦,𝐷

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpox

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1920	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		cbvmpox.1	. . . . . 6 ⊢ Ⅎ𝑧𝐵
3	2	nfcri 2886	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
4	1, 3	nfan 1905	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
5		cbvmpox.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
6	5	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
7	4, 6	nfan 1905	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
8		nfv 1920	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
9		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑤𝐵
10	9	nfcri 2886	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
11	8, 10	nfan 1905	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		cbvmpox.4	. . . . 5 ⊢ Ⅎ𝑤𝐶
13	12	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
14	11, 13	nfan 1905	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
15		nfv 1920	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
16		cbvmpox.2	. . . . . 6 ⊢ Ⅎ𝑥𝐷
17	16	nfcri 2886	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
18	15, 17	nfan 1905	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
19		cbvmpox.5	. . . . 5 ⊢ Ⅎ𝑥𝐸
20	19	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
21	18, 20	nfan 1905	. . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
22		nfv 1920	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
23		cbvmpox.6	. . . . 5 ⊢ Ⅎ𝑦𝐸
24	23	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
25	22, 24	nfan 1905	. . 3 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
26		eleq1w 2815	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 484	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		cbvmpox.7	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)
29	28	eleq2d 2818	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐷))
30		eleq1w 2815	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐷 ↔ 𝑤 ∈ 𝐷))
31	29, 30	sylan9bb 513	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐷))
32	27, 31	anbi12d 634	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)))
33		cbvmpox.8	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸)
34	33	eqeq2d 2749	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
35	32, 34	anbi12d 634	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)))
36	7, 14, 21, 25, 35	cbvoprab12 7251	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
37		df-mpo 7169	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
38		df-mpo 7169	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
39	36, 37, 38	3eqtr4i 2771	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  Ⅎwnfc 2879  {coprab 7165   ∈ cmpo 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-opab 5090  df-oprab 7168  df-mpo 7169

This theorem is referenced by:  cbvmpo  7256  mpomptsx  7780  dmmpossx  7782  gsumcom2  19207  ptcmpg  22801