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Theorem nfmpo 5806
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpo.1  |-  F/_ z A
nfmpo.2  |-  F/_ z B
nfmpo.3  |-  F/_ z C
Assertion
Ref Expression
nfmpo  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    A( x, y, z)    B( x, y, z)    C( x, y, z)

Proof of Theorem nfmpo
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-mpo 5745 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
2 nfmpo.1 . . . . . 6  |-  F/_ z A
32nfcri 2250 . . . . 5  |-  F/ z  x  e.  A
4 nfmpo.2 . . . . . 6  |-  F/_ z B
54nfcri 2250 . . . . 5  |-  F/ z  y  e.  B
63, 5nfan 1527 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
7 nfmpo.3 . . . . 5  |-  F/_ z C
87nfeq2 2268 . . . 4  |-  F/ z  w  =  C
96, 8nfan 1527 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C )
109nfoprab 5789 . 2  |-  F/_ z { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
111, 10nfcxfr 2253 1  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314    e. wcel 1463   F/_wnfc 2243   {coprab 5741    e. cmpo 5742
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  nfof  5953  nfseq  10179
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