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	|- F/_yB
cbvmpt.2	|- F/_xC
cbvmpt.3	|- (x = y -> B = C)

Assertion

Ref	Expression
cbvmpt	|- (x e. A |-> B) = (y e. 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 |- F/w(x e. A /\ z = B)
2		nfv 1619	. . . . 5 |- F/x w e. A
3		nfs1v 2106	. . . . 5 |- F/x[w / x]z = B
4	2, 3	nfan 1824	. . . 4 |- F/x(w e. A /\ [w / x]z = B)
5		eleq1 2413	. . . . 5 |- (x = w -> (x e. A <-> w e. A))
6		sbequ12 1919	. . . . 5 |- (x = w -> (z = B <-> [w / x]z = B))
7	5, 6	anbi12d 691	. . . 4 |- (x = w -> ((x e. A /\ z = B) <-> (w e. A /\ [w / x]z = B)))
8	1, 4, 7	cbvopab1 4633	. . 3 |- {<.x, z>. | (x e. A /\ z = B)} = {<.w, z>. | (w e. A /\ [w / x]z = B)}
9		nfv 1619	. . . . 5 |- F/y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_yB
11	10	nfeq2 2501	. . . . . 6 |- F/y z = B
12	11	nfsb 2109	. . . . 5 |- F/y[w / x]z = B
13	9, 12	nfan 1824	. . . 4 |- F/y(w e. A /\ [w / x]z = B)
14		nfv 1619	. . . 4 |- F/w(y e. A /\ z = C)
15		eleq1 2413	. . . . 5 |- (w = y -> (w e. A <-> y e. A))
16		sbequ 2060	. . . . . 6 |- (w = y -> ([w / x]z = B <-> [y / x]z = B))
17		cbvmpt.2	. . . . . . . 8 |- F/_xC
18	17	nfeq2 2501	. . . . . . 7 |- F/x 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 e. A /\ [w / x]z = B) <-> (y e. A /\ z = C)))
24	13, 14, 23	cbvopab1 4633	. . 3 |- {<.w, z>. | (w e. A /\ [w / x]z = B)} = {<.y, z>. | (y e. A /\ z = C)}
25	8, 24	eqtri 2373	. 2 |- {<.x, z>. | (x e. A /\ z = B)} = {<.y, z>. | (y e. A /\ z = C)}
26		df-mpt 5653	. 2 |- (x e. A |-> B) = {<.x, z>. | (x e. A /\ z = B)}
27		df-mpt 5653	. 2 |- (y e. A |-> C) = {<.y, z>. | (y e. A /\ z = C)}
28	25, 26, 27	3eqtr4i 2383	1 |- (x e. A |-> B) = (y e. A |-> C)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  [wsb 1648   e. wcel 1710   F/_wnfc 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