Theorem cbvmpt2x2 42114

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6734 allows 𝐴 to be a function of 𝑦, analogous to cbvmpt2x 6733. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
cbvmpt2x2.1	⊢ Ⅎ𝑧𝐴
cbvmpt2x2.2	⊢ Ⅎ𝑦𝐷
cbvmpt2x2.3	⊢ Ⅎ𝑧𝐶
cbvmpt2x2.4	⊢ Ⅎ𝑤𝐶
cbvmpt2x2.5	⊢ Ⅎ𝑥𝐸
cbvmpt2x2.6	⊢ Ⅎ𝑦𝐸
cbvmpt2x2.7	⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷)
cbvmpt2x2.8	⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpt2x2	⊢ (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 cbvmpt2x2

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
2		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
3	1, 2	nfan 1828	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
4		cbvmpt2x2.4	. . . . 5 ⊢ Ⅎ𝑤𝐶
5	4	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
6	3, 5	nfan 1828	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
7		cbvmpt2x2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
8	7	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
9		nfv 1843	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
10	8, 9	nfan 1828	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
11		cbvmpt2x2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
12	11	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
13	10, 12	nfan 1828	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
14		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
15		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
16	14, 15	nfan 1828	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)
17		cbvmpt2x2.5	. . . . 5 ⊢ Ⅎ𝑥𝐸
18	17	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
19	16, 18	nfan 1828	. . 3 ⊢ Ⅎ𝑥((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)
20		cbvmpt2x2.2	. . . . . 6 ⊢ Ⅎ𝑦𝐷
21	20	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐷
22		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
23	21, 22	nfan 1828	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)
24		cbvmpt2x2.6	. . . . 5 ⊢ Ⅎ𝑦𝐸
25	24	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
26	23, 25	nfan 1828	. . 3 ⊢ Ⅎ𝑦((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)
27		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
28		cbvmpt2x2.7	. . . . . . 7 ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷)
29	28	eleq2d 2687	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐷))
30	27, 29	sylan9bb 736	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐷))
31		simpr 477	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
32	31	eleq1d 2686	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
33	30, 32	anbi12d 747	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵)))
34		cbvmpt2x2.8	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸)
35	34	ancoms 469	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → 𝐶 = 𝐸)
36	35	eqeq2d 2632	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
37	33, 36	anbi12d 747	. . 3 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
38	6, 13, 19, 26, 37	cbvoprab12 6729	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
39		df-mpt2 6655	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
40		df-mpt2 6655	. 2 ⊢ (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑤, 𝑧⟩, 𝑢⟩ ∣ ((𝑤 ∈ 𝐷 ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
41	38, 39, 40	3eqtr4i 2654	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  {coprab 6651   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dmmpt2ssx2  42115