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Theorem nfmpt 3890
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt.1  |-  F/_ x A
nfmpt.2  |-  F/_ x B
Assertion
Ref Expression
nfmpt  |-  F/_ x
( y  e.  A  |->  B )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem nfmpt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3861 . 2  |-  ( y  e.  A  |->  B )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  B ) }
2 nfmpt.1 . . . . 5  |-  F/_ x A
32nfcri 2217 . . . 4  |-  F/ x  y  e.  A
4 nfmpt.2 . . . . 5  |-  F/_ x B
54nfeq2 2234 . . . 4  |-  F/ x  z  =  B
63, 5nfan 1498 . . 3  |-  F/ x
( y  e.  A  /\  z  =  B
)
76nfopab 3866 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  =  B ) }
81, 7nfcxfr 2220 1  |-  F/_ x
( y  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   F/_wnfc 2210   {copab 3858    |-> cmpt 3859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-opab 3860  df-mpt 3861
This theorem is referenced by:  nffrec  6066  nfsum1  10412  nfsum  10413
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