Theorem cbvmpt22 38305

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
cbvmpt22.1	⊢ Ⅎ𝑦𝐴
cbvmpt22.2	⊢ Ⅎ𝑤𝐴
cbvmpt22.3	⊢ Ⅎ𝑤𝐶
cbvmpt22.4	⊢ Ⅎ𝑦𝐸
cbvmpt22.5	⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpt22	⊢ (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 cbvmpt22

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑤𝑥
2		cbvmpt22.2	. . . . . 6 ⊢ Ⅎ𝑤𝐴
3	1, 2	nfel 2763	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
4		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑤𝐵
5	4	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
6	3, 5	nfan 1816	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		cbvmpt22.3	. . . . 5 ⊢ Ⅎ𝑤𝐶
8	7	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
9	6, 8	nfan 1816	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
10		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑦𝑥
11		cbvmpt22.1	. . . . . 6 ⊢ Ⅎ𝑦𝐴
12	10, 11	nfel 2763	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
13		nfv 1830	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐵
14	12, 13	nfan 1816	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
15		cbvmpt22.4	. . . . 5 ⊢ Ⅎ𝑦𝐸
16	15	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
17	14, 16	nfan 1816	. . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)
18		eleq1 2676	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
19	18	anbi2d 736	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
20		cbvmpt22.5	. . . . 5 ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)
21	20	eqeq2d 2620	. . . 4 ⊢ (𝑦 = 𝑤 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
22	19, 21	anbi12d 743	. . 3 ⊢ (𝑦 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)))
23	9, 17, 22	cbvoprab2 6626	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
24		df-mpt2 6554	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
25		df-mpt2 6554	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)}
26	23, 24, 25	3eqtr4i 2642	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  smflimlem4  39660