Theorem nfmpt1 4137

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 4107	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4113	. 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 4104   |-> cmpt 4105

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 4106  df-mpt 4107

This theorem is referenced by:  nffvmpt1  5587  fvmptss2  5654  fvmptssdm  5664  fvmptdf  5667  mpteqb  5670  fvmptf  5672  ralrnmpt  5722  rexrnmpt  5723  f1ompt  5731  f1mpt  5840  fliftfun  5865  dom2lem  6863  mapxpen  6945  mkvprop  7260  cc3  7380  nfcprod1  11865  cnmpt11  14755  lgseisenlem2  15548