Theorem cbvmpo2 41733

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 2943	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
3		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑤𝐵
4	3	nfcri 2943	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
5	2, 4	nfan 1900	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
6		cbvmpo2.3	. . . . 5 ⊢ Ⅎ𝑤𝐶
7	6	nfeq2 2972	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
8	5, 7	nfan 1900	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
9		cbvmpo2.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
10	9	nfcri 2943	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
11		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐵
12	10, 11	nfan 1900	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
13		cbvmpo2.4	. . . . 5 ⊢ Ⅎ𝑦𝐸
14	13	nfeq2 2972	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
15	12, 14	nfan 1900	. . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)
16		eleq1w 2872	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
17	16	anbi2d 631	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
18		cbvmpo2.5	. . . . 5 ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)
19	18	eqeq2d 2809	. . . 4 ⊢ (𝑦 = 𝑤 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
20	17, 19	anbi12d 633	. . 3 ⊢ (𝑦 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
21	8, 15, 20	cbvoprab2 7221	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
22		df-mpo 7140	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
23		df-mpo 7140	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
24	21, 22, 23	3eqtr4i 2831	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  {coprab 7136   ∈ cmpo 7137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  smflimlem4  43407