Theorem cmpbva 10454

Description: Rule to change a bound variable int the "maps to" notation.

Hypothesis

Ref	Expression
cmpbva.1	|- F = (x e. A |-> C)

Assertion

Ref	Expression
cmpbva	|- F = (y e. A |-> C)

Distinct variable groups:   x,A,y   x,C,y

Proof of Theorem cmpbva

Step	Hyp	Ref	Expression
1		cmpbva.1	. . 3 |- F = (x e. A |-> C)
2		df-mpt 4079	. . 3 |- (x e. A |-> C) = {<.x, w>. | (x e. A /\ w = C)}
3		eleq1 1537	. . . . 5 |- (x = y -> (x e. A <-> y e. A))
4		eqeq1 1484	. . . . 5 |- (w = z -> (w = C <-> z = C))
5	3, 4	bi2anan9 634	. . . 4 |- ((x = y /\ w = z) -> ((x e. A /\ w = C) <-> (y e. A /\ z = C)))
6	5	cbvopabv 2678	. . 3 |- {<.x, w>. | (x e. A /\ w = C)} = {<.y, z>. | (y e. A /\ z = C)}
7	1, 2, 6	3eqtr 1502	. 2 |- F = {<.y, z>. | (y e. A /\ z = C)}
8		df-mpt 4079	. 2 |- (y e. A |-> C) = {<.y, z>. | (y e. A /\ z = C)}
9	7, 8	eqtr4 1501	1 |- F = (y e. A |-> C)

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  {copab 2671   e. cmpt 4077

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-mpt 4079

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