Theorem cbvmpo1 45103

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 1914	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		cbvmpo1.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
3	2	nfcri 2897	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
4	1, 3	nfan 1899	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
5		cbvmpo1.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
6	5	nfeq2 2923	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
7	4, 6	nfan 1899	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
8		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
9		cbvmpo1.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10	9	nfcri 2897	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
11	8, 10	nfan 1899	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		cbvmpo1.4	. . . . 5 ⊢ Ⅎ𝑥𝐸
13	12	nfeq2 2923	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
14	11, 13	nfan 1899	. . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)
15		eleq1w 2824	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
16	15	anbi1d 631	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
17		cbvmpo1.5	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
18	17	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
19	16, 18	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
20	7, 14, 19	cbvoprab1 7520	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
21		df-mpo 7436	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
22		df-mpo 7436	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
23	20, 21, 22	3eqtr4i 2775	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  {coprab 7432   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-oprab 7435  df-mpo 7436

This theorem is referenced by: (None)