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Theorem nfmpt 4023
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 3994 . 2  |-  ( y  e.  A  |->  B )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  B ) }
2 nfmpt.1 . . . . 5  |-  F/_ x A
32nfcri 2275 . . . 4  |-  F/ x  y  e.  A
4 nfmpt.2 . . . . 5  |-  F/_ x B
54nfeq2 2293 . . . 4  |-  F/ x  z  =  B
63, 5nfan 1544 . . 3  |-  F/ x
( y  e.  A  /\  z  =  B
)
76nfopab 3999 . 2  |-  F/_ x { <. y ,  z
>.  |  ( y  e.  A  /\  z  =  B ) }
81, 7nfcxfr 2278 1  |-  F/_ x
( y  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331    e. wcel 1480   F/_wnfc 2268   {copab 3991    |-> cmpt 3992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-opab 3993  df-mpt 3994
This theorem is referenced by:  nfof  5990  nffrec  6296  mapxpen  6745  nfsum1  11149  nfsum  11150  nfcprod1  11347  nfcprod  11348  ctiunct  11976  fsumcncntop  12751  limcmpted  12827
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