Theorem cbvmpt2x 6631

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6632 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
cbvmpt2x.1	⊢ Ⅎ𝑧𝐵
cbvmpt2x.2	⊢ Ⅎ𝑥𝐷
cbvmpt2x.3	⊢ Ⅎ𝑧𝐶
cbvmpt2x.4	⊢ Ⅎ𝑤𝐶
cbvmpt2x.5	⊢ Ⅎ𝑥𝐸
cbvmpt2x.6	⊢ Ⅎ𝑦𝐸
cbvmpt2x.7	⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)
cbvmpt2x.8	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸)

Assertion

Ref	Expression
cbvmpt2x	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵   𝑦,𝐷

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

Proof of Theorem cbvmpt2x

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1830	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		cbvmpt2x.1	. . . . . 6 ⊢ Ⅎ𝑧𝐵
3	2	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
4	1, 3	nfan 1816	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
5		cbvmpt2x.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
6	5	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
7	4, 6	nfan 1816	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
8		nfv 1830	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
9		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑤𝐵
10	9	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
11	8, 10	nfan 1816	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		cbvmpt2x.4	. . . . 5 ⊢ Ⅎ𝑤𝐶
13	12	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
14	11, 13	nfan 1816	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
15		nfv 1830	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
16		cbvmpt2x.2	. . . . . 6 ⊢ Ⅎ𝑥𝐷
17	16	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
18	15, 17	nfan 1816	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
19		cbvmpt2x.5	. . . . 5 ⊢ Ⅎ𝑥𝐸
20	19	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
21	18, 20	nfan 1816	. . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
22		nfv 1830	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
23		cbvmpt2x.6	. . . . 5 ⊢ Ⅎ𝑦𝐸
24	23	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
25	22, 24	nfan 1816	. . 3 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
26		eleq1 2676	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 480	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		cbvmpt2x.7	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)
29	28	eleq2d 2673	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐷))
30		eleq1 2676	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐷 ↔ 𝑤 ∈ 𝐷))
31	29, 30	sylan9bb 732	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐷))
32	27, 31	anbi12d 743	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)))
33		cbvmpt2x.8	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸)
34	33	eqeq2d 2620	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
35	32, 34	anbi12d 743	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)))
36	7, 14, 21, 25, 35	cbvoprab12 6627	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
37		df-mpt2 6554	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
38		df-mpt2 6554	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
39	36, 37, 38	3eqtr4i 2642	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)

Syntax hints:   → wi 4   ↔ wb 195   ∧ 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-opab 4644  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  cbvmpt2  6632  mpt2mptsx  7122  dmmpt2ssx  7124  gsumcom2  18197  ptcmpg  21671