Theorem nfmpt1 4029

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 3999	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4005	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2279	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 1481   F/_wnfc 2269   {copab 3996   |-> cmpt 3997

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-opab 3998  df-mpt 3999

This theorem is referenced by:  nffvmpt1  5440  fvmptss2  5504  fvmptssdm  5513  fvmptdf  5516  mpteqb  5519  fvmptf  5521  ralrnmpt  5570  rexrnmpt  5571  f1ompt  5579  f1mpt  5680  fliftfun  5705  dom2lem  6674  mapxpen  6750  mkvprop  7040  cc3  7100  nfcprod1  11355  cnmpt11  12491