Theorem cbvmpo1 42170

Description: Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
cbvmpo1.1	⊢ Ⅎ𝑥𝐵
cbvmpo1.2	⊢ Ⅎ𝑧𝐵
cbvmpo1.3	⊢ Ⅎ𝑧𝐶
cbvmpo1.4	⊢ Ⅎ𝑥𝐸
cbvmpo1.5	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpo1	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝑦,𝑧

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

Proof of Theorem cbvmpo1

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1920	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		cbvmpo1.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
3	2	nfcri 2886	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
4	1, 3	nfan 1905	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
5		cbvmpo1.3	. . . . 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 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
9		cbvmpo1.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10	9	nfcri 2886	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
11	8, 10	nfan 1905	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		cbvmpo1.4	. . . . 5 ⊢ Ⅎ𝑥𝐸
13	12	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
14	11, 13	nfan 1905	. . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)
15		eleq1w 2815	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
16	15	anbi1d 633	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17		cbvmpo1.5	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
18	17	eqeq2d 2749	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
19	16, 18	anbi12d 634	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
20	7, 14, 19	cbvoprab1 7249	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
21		df-mpo 7169	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
22		df-mpo 7169	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
23	20, 21, 22	3eqtr4i 2771	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   → wi 4   ∧ 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: (None)