Theorem nfmpt1 4153

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 4123	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4129	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2347	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   F/_wnfc 2337   {copab 4120   |-> cmpt 4121

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-opab 4122  df-mpt 4123

This theorem is referenced by:  nffvmpt1  5610  fvmptss2  5677  fvmptssdm  5687  fvmptdf  5690  mpteqb  5693  fvmptf  5695  ralrnmpt  5745  rexrnmpt  5746  f1ompt  5754  f1mpt  5863  fliftfun  5888  dom2lem  6886  mapxpen  6970  mkvprop  7286  cc3  7415  nfcprod1  11980  cnmpt11  14870  lgseisenlem2  15663