Theorem nfmpt1 4126

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 4096	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4102	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2336	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   F/_wnfc 2326   {copab 4093   |-> cmpt 4094

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-opab 4095  df-mpt 4096

This theorem is referenced by:  nffvmpt1  5569  fvmptss2  5636  fvmptssdm  5646  fvmptdf  5649  mpteqb  5652  fvmptf  5654  ralrnmpt  5704  rexrnmpt  5705  f1ompt  5713  f1mpt  5818  fliftfun  5843  dom2lem  6831  mapxpen  6909  mkvprop  7224  cc3  7335  nfcprod1  11719  cnmpt11  14519  lgseisenlem2  15312