Theorem nfmpt1 4138

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 4108	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4114	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2345	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   F/_wnfc 2335   {copab 4105   |-> cmpt 4106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-opab 4107  df-mpt 4108

This theorem is referenced by:  nffvmpt1  5589  fvmptss2  5656  fvmptssdm  5666  fvmptdf  5669  mpteqb  5672  fvmptf  5674  ralrnmpt  5724  rexrnmpt  5725  f1ompt  5733  f1mpt  5842  fliftfun  5867  dom2lem  6865  mapxpen  6947  mkvprop  7262  cc3  7382  nfcprod1  11898  cnmpt11  14788  lgseisenlem2  15581