Theorem nfmpo1 5993

Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Assertion

Ref	Expression
nfmpo1	|- F/_ x ( x e. A , y e. B |-> C )

Proof of Theorem nfmpo1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 5930	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		nfoprab1 5975	. 2 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
3	1, 2	nfcxfr 2336	1 |- F/_ x ( x e. A , y e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   F/_wnfc 2326   {coprab 5926   e. cmpo 5927

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-oprab 5929  df-mpo 5930

This theorem is referenced by:  ovmpos  6050  ov2gf  6051  ovmpodxf  6052  ovmpodf  6058  ovmpodv2  6060  xpcomco  6894  mapxpen  6918  cnmpt21  14611  cnmpt2t  14613  cnmptcom  14618