Theorem nfmpt1 4182

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 4152	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
2		nfopab1 4158	. 2 |- F/_ x { <. x , z >. | ( x e. A /\ z = B ) }
3	1, 2	nfcxfr 2371	1 |- F/_ x ( x e. A |-> B )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   F/_wnfc 2361   {copab 4149   |-> cmpt 4150

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-opab 4151  df-mpt 4152

This theorem is referenced by:  nffvmpt1  5651  fvmptss2  5722  fvmptssdm  5732  fvmptdf  5735  mpteqb  5738  fvmptf  5740  ralrnmpt  5790  rexrnmpt  5791  f1ompt  5799  f1mpt  5915  fliftfun  5940  dom2lem  6948  mapxpen  7037  mkvprop  7360  cc3  7490  nfcprod1  12136  cnmpt11  15034  lgseisenlem2  15827