Theorem nfmpt1 4057

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 4027	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4033	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2296	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1335   e. wcel 2128   F/_wnfc 2286   {copab 4024   |-> cmpt 4025

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-opab 4026  df-mpt 4027

This theorem is referenced by:  nffvmpt1  5478  fvmptss2  5542  fvmptssdm  5551  fvmptdf  5554  mpteqb  5557  fvmptf  5559  ralrnmpt  5608  rexrnmpt  5609  f1ompt  5617  f1mpt  5718  fliftfun  5743  dom2lem  6714  mapxpen  6790  mkvprop  7096  cc3  7183  nfcprod1  11446  cnmpt11  12670