Theorem nfmpt 4020

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 3991	. 2 |- ( y e. A |-> B ) = { <. y , z >. | ( y e. A /\ z = B ) }
2		nfmpt.1	. . . . 5 |- F/_ x A
3	2	nfcri 2275	. . . 4 |- F/ x y e. A
4		nfmpt.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2293	. . . 4 |- F/ x z = B
6	3, 5	nfan 1544	. . 3 |- F/ x ( y e. A /\ z = B )
7	6	nfopab 3996	. 2 |- F/_ x { <. y , z >. | ( y e. A /\ z = B ) }
8	1, 7	nfcxfr 2278	1 |- F/_ x ( y e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   F/_wnfc 2268   {copab 3988   |-> cmpt 3989

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-opab 3990  df-mpt 3991

This theorem is referenced by:  nfof  5987  nffrec  6293  mapxpen  6742  nfsum1  11125  nfsum  11126  nfcprod1  11323  nfcprod  11324  ctiunct  11953  fsumcncntop  12725  limcmpted  12801