Theorem nfmpt1 4127

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 4097	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4103	. 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 4094   |-> cmpt 4095

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 4096  df-mpt 4097

This theorem is referenced by:  nffvmpt1  5570  fvmptss2  5637  fvmptssdm  5647  fvmptdf  5650  mpteqb  5653  fvmptf  5655  ralrnmpt  5705  rexrnmpt  5706  f1ompt  5714  f1mpt  5819  fliftfun  5844  dom2lem  6833  mapxpen  6911  mkvprop  7226  cc3  7338  nfcprod1  11722  cnmpt11  14545  lgseisenlem2  15338