Theorem cbvmpo2 46385

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

Hypotheses

Ref	Expression
cbvmpo2.1	⊢ Ⅎ𝑦𝐴
cbvmpo2.2	⊢ Ⅎ𝑤𝐴
cbvmpo2.3	⊢ Ⅎ𝑤𝐶
cbvmpo2.4	⊢ Ⅎ𝑦𝐸
cbvmpo2.5	⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpo2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)

Distinct variable groups:   𝑤,𝐵,𝑦   𝑥,𝑤,𝑦

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

Proof of Theorem cbvmpo2

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmpo2.2	. . . . . 6 ⊢ Ⅎ𝑤𝐴
2	1	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
3		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑤𝐵
4	3	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
5	2, 4	nfan 1934	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
6		cbvmpo2.3	. . . . 5 ⊢ Ⅎ𝑤𝐶
7	6	nfeq2 2943	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
8	5, 7	nfan 1934	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
9		cbvmpo2.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
10	9	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
11		nfv 1949	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐵
12	10, 11	nfan 1934	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
13		cbvmpo2.4	. . . . 5 ⊢ Ⅎ𝑦𝐸
14	13	nfeq2 2943	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
15	12, 14	nfan 1934	. . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)
16		eleq1w 2847	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
17	16	anbi2d 642	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
18		cbvmpo2.5	. . . . 5 ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)
19	18	eqeq2d 2774	. . . 4 ⊢ (𝑦 = 𝑤 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
20	17, 19	anbi12d 644	. . 3 ⊢ (𝑦 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
21	8, 15, 20	cbvoprab2 7539	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
22		df-mpo 7456	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
23		df-mpo 7456	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
24	21, 22, 23	3eqtr4i 2796	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  {coprab 7452   ∈ cmpo 7453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-oprab 7455  df-mpo 7456

This theorem is used by:  smflimlem4  48057