Theorem cbvmpox2 45671

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7369 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7368. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
cbvmpox2.1	⊢ Ⅎ𝑧𝐴
cbvmpox2.2	⊢ Ⅎ𝑦𝐷
cbvmpox2.3	⊢ Ⅎ𝑧𝐶
cbvmpox2.4	⊢ Ⅎ𝑤𝐶
cbvmpox2.5	⊢ Ⅎ𝑥𝐸
cbvmpox2.6	⊢ Ⅎ𝑦𝐸
cbvmpox2.7	⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷)
cbvmpox2.8	⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpox2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸)

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

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

Proof of Theorem cbvmpox2

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1917	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
2		nfv 1917	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
3	1, 2	nfan 1902	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
4		cbvmpox2.4	. . . . 5 ⊢ Ⅎ𝑤𝐶
5	4	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
6	3, 5	nfan 1902	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
7		cbvmpox2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
8	7	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
9		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
10	8, 9	nfan 1902	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
11		cbvmpox2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
12	11	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
13	10, 12	nfan 1902	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
14		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
15		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
16	14, 15	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)
17		cbvmpox2.5	. . . . 5 ⊢ Ⅎ𝑥𝐸
18	17	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
19	16, 18	nfan 1902	. . 3 ⊢ Ⅎ𝑥((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)
20		cbvmpox2.2	. . . . . 6 ⊢ Ⅎ𝑦𝐷
21	20	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐷
22		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
23	21, 22	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)
24		cbvmpox2.6	. . . . 5 ⊢ Ⅎ𝑦𝐸
25	24	nfeq2 2924	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
26	23, 25	nfan 1902	. . 3 ⊢ Ⅎ𝑦((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)
27		eleq1w 2821	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
28		cbvmpox2.7	. . . . . . 7 ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷)
29	28	eleq2d 2824	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐷))
30	27, 29	sylan9bb 510	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐷))
31		simpr 485	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
32	31	eleq1d 2823	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
33	30, 32	anbi12d 631	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)))
34		cbvmpox2.8	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸)
35	34	ancoms 459	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → 𝐶 = 𝐸)
36	35	eqeq2d 2749	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
37	33, 36	anbi12d 631	. . 3 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
38	6, 13, 19, 26, 37	cbvoprab12 7364	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
39		df-mpo 7280	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
40		df-mpo 7280	. 2 ⊢ (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
41	38, 39, 40	3eqtr4i 2776	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  {coprab 7276   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  dmmpossx2  45672