Theorem nfmpt1 4021

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)

Assertion

Ref	Expression
nfmpt1	|- F/_ x ( x e. A |-> B )

Proof of Theorem nfmpt1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 3991	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 3997	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2278	1 |- F/_ x ( x 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-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:  nffvmpt1  5432  fvmptss2  5496  fvmptssdm  5505  fvmptdf  5508  mpteqb  5511  fvmptf  5513  ralrnmpt  5562  rexrnmpt  5563  f1ompt  5571  f1mpt  5672  fliftfun  5697  dom2lem  6666  mapxpen  6742  mkvprop  7032  nfcprod1  11323  cnmpt11  12452