Theorem cbvmpt 5677

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)

Hypotheses

Ref	Expression
cbvmpt.1	⊢ ℲyB
cbvmpt.2	⊢ ℲxC
cbvmpt.3	⊢ (x = y → B = C)

Assertion

Ref	Expression
cbvmpt	⊢ (x ∈ A ↦ B) = (y ∈ A ↦ C)

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   B(x,y)   C(x,y)

Proof of Theorem cbvmpt

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎw(x ∈ A ∧ z = B)
2		nfv 1619	. . . . 5 ⊢ Ⅎx w ∈ A
3		nfs1v 2106	. . . . 5 ⊢ Ⅎx[w / x]z = B
4	2, 3	nfan 1824	. . . 4 ⊢ Ⅎx(w ∈ A ∧ [w / x]z = B)
5		eleq1 2413	. . . . 5 ⊢ (x = w → (x ∈ A ↔ w ∈ A))
6		sbequ12 1919	. . . . 5 ⊢ (x = w → (z = B ↔ [w / x]z = B))
7	5, 6	anbi12d 691	. . . 4 ⊢ (x = w → ((x ∈ A ∧ z = B) ↔ (w ∈ A ∧ [w / x]z = B)))
8	1, 4, 7	cbvopab1 4633	. . 3 ⊢ {⟨x, z⟩ ∣ (x ∈ A ∧ z = B)} = {⟨w, z⟩ ∣ (w ∈ A ∧ [w / x]z = B)}
9		nfv 1619	. . . . 5 ⊢ Ⅎy w ∈ A
10		cbvmpt.1	. . . . . . 7 ⊢ ℲyB
11	10	nfeq2 2501	. . . . . 6 ⊢ Ⅎy z = B
12	11	nfsb 2109	. . . . 5 ⊢ Ⅎy[w / x]z = B
13	9, 12	nfan 1824	. . . 4 ⊢ Ⅎy(w ∈ A ∧ [w / x]z = B)
14		nfv 1619	. . . 4 ⊢ Ⅎw(y ∈ A ∧ z = C)
15		eleq1 2413	. . . . 5 ⊢ (w = y → (w ∈ A ↔ y ∈ A))
16		sbequ 2060	. . . . . 6 ⊢ (w = y → ([w / x]z = B ↔ [y / x]z = B))
17		cbvmpt.2	. . . . . . . 8 ⊢ ℲxC
18	17	nfeq2 2501	. . . . . . 7 ⊢ Ⅎx z = C
19		cbvmpt.3	. . . . . . . 8 ⊢ (x = y → B = C)
20	19	eqeq2d 2364	. . . . . . 7 ⊢ (x = y → (z = B ↔ z = C))
21	18, 20	sbie 2038	. . . . . 6 ⊢ ([y / x]z = B ↔ z = C)
22	16, 21	syl6bb 252	. . . . 5 ⊢ (w = y → ([w / x]z = B ↔ z = C))
23	15, 22	anbi12d 691	. . . 4 ⊢ (w = y → ((w ∈ A ∧ [w / x]z = B) ↔ (y ∈ A ∧ z = C)))
24	13, 14, 23	cbvopab1 4633	. . 3 ⊢ {⟨w, z⟩ ∣ (w ∈ A ∧ [w / x]z = B)} = {⟨y, z⟩ ∣ (y ∈ A ∧ z = C)}
25	8, 24	eqtri 2373	. 2 ⊢ {⟨x, z⟩ ∣ (x ∈ A ∧ z = B)} = {⟨y, z⟩ ∣ (y ∈ A ∧ z = C)}
26		df-mpt 5653	. 2 ⊢ (x ∈ A ↦ B) = {⟨x, z⟩ ∣ (x ∈ A ∧ z = B)}
27		df-mpt 5653	. 2 ⊢ (y ∈ A ↦ C) = {⟨y, z⟩ ∣ (y ∈ A ∧ z = C)}
28	25, 26, 27	3eqtr4i 2383	1 ⊢ (x ∈ A ↦ B) = (y ∈ A ↦ C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2477  {copab 4623   ↦ cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-mpt 5653

This theorem is referenced by:  cbvmptv  5678  fvmpts  5702  fvmpt2i  5704  fvmptex  5722