Theorem nfmpt1 4177

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 4147	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4153	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2369	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   F/_wnfc 2359   {copab 4144   |-> cmpt 4145

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-opab 4146  df-mpt 4147

This theorem is referenced by:  nffvmpt1  5638  fvmptss2  5709  fvmptssdm  5719  fvmptdf  5722  mpteqb  5725  fvmptf  5727  ralrnmpt  5777  rexrnmpt  5778  f1ompt  5786  f1mpt  5895  fliftfun  5920  dom2lem  6923  mapxpen  7009  mkvprop  7325  cc3  7454  nfcprod1  12065  cnmpt11  14957  lgseisenlem2  15750