Theorem nfmpt 4074

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Hypotheses

Ref	Expression
nfmpt.1	|- F/_ x A
nfmpt.2	|- F/_ x B

Assertion

Ref	Expression
nfmpt	|- F/_ x ( y e. A |-> B )

Distinct variable group:   x, y

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

Proof of Theorem nfmpt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4045	. 2 |- ( y e. A |-> B ) = { <. y , z >. | ( y e. A /\ z = B ) }
2		nfmpt.1	. . . . 5 |- F/_ x A
3	2	nfcri 2302	. . . 4 |- F/ x y e. A
4		nfmpt.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2320	. . . 4 |- F/ x z = B
6	3, 5	nfan 1553	. . 3 |- F/ x ( y e. A /\ z = B )
7	6	nfopab 4050	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z = B ) }
8	1, 7	nfcxfr 2305	1 |- F/_ x ( y e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   F/_wnfc 2295   {copab 4042   |-> cmpt 4043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-opab 4044  df-mpt 4045

This theorem is referenced by:  nfof  6055  nffrec  6364  mapxpen  6814  nfsum1  11297  nfsum  11298  nfcprod1  11495  nfcprod  11496  ctiunct  12373  fsumcncntop  13206  limcmpted  13282