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Theorem nfmpt2 6134
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt2.1  |-  F/_ z A
nfmpt2.2  |-  F/_ z B
nfmpt2.3  |-  F/_ z C
Assertion
Ref Expression
nfmpt2  |-  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 nfmpt2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-mpt2 6078 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
2 nfmpt2.1 . . . . . 6  |-  F/_ z A
32nfcri 2565 . . . . 5  |-  F/ z  x  e.  A
4 nfmpt2.2 . . . . . 6  |-  F/_ z B
54nfcri 2565 . . . . 5  |-  F/ z  y  e.  B
63, 5nfan 1846 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
7 nfmpt2.3 . . . . 5  |-  F/_ z C
87nfeq2 2582 . . . 4  |-  F/ z  w  =  C
96, 8nfan 1846 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C )
109nfoprab 6118 . 2  |-  F/_ z { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
111, 10nfcxfr 2568 1  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   F/_wnfc 2558   {coprab 6074    e. cmpt2 6075
This theorem is referenced by:  nfseq  11325  ptbasfi  17605  sdclem1  26438  fmuldfeqlem1  27679  stoweidlem51  27767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-oprab 6077  df-mpt2 6078
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