Theorem cbvmpt2x 6733

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6734 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 1843	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		cbvmpt2x.1	. . . . . 6 ⊢ Ⅎ𝑧𝐵
3	2	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
4	1, 3	nfan 1828	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
5		cbvmpt2x.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
6	5	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶
7	4, 6	nfan 1828	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
8		nfv 1843	. . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
9		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑤𝐵
10	9	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
11	8, 10	nfan 1828	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		cbvmpt2x.4	. . . . 5 ⊢ Ⅎ𝑤𝐶
13	12	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶
14	11, 13	nfan 1828	. . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)
15		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
16		cbvmpt2x.2	. . . . . 6 ⊢ Ⅎ𝑥𝐷
17	16	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
18	15, 17	nfan 1828	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
19		cbvmpt2x.5	. . . . 5 ⊢ Ⅎ𝑥𝐸
20	19	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸
21	18, 20	nfan 1828	. . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
22		nfv 1843	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷)
23		cbvmpt2x.6	. . . . 5 ⊢ Ⅎ𝑦𝐸
24	23	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸
25	22, 24	nfan 1828	. . 3 ⊢ Ⅎ𝑦((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)
26		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
27	26	adantr 481	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
28		cbvmpt2x.7	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷)
29	28	eleq2d 2687	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐷))
30		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐷 ↔ 𝑤 ∈ 𝐷))
31	29, 30	sylan9bb 736	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐷))
32	27, 31	anbi12d 747	. . . 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 2632	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸))
35	32, 34	anbi12d 747	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)))
36	7, 14, 21, 25, 35	cbvoprab12 6729	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
37		df-mpt2 6655	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)}
38		df-mpt2 6655	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 = 𝐸)}
39	36, 37, 38	3eqtr4i 2654	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  {coprab 6651   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  cbvmpt2  6734  mpt2mptsx  7233  dmmpt2ssx  7235  gsumcom2  18374  ptcmpg  21861