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Theorem nfmpt2 5916
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 5863 . 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 2413 . . . . 5  |-  F/ z  x  e.  A
4 nfmpt2.2 . . . . . 6  |-  F/_ z B
54nfcri 2413 . . . . 5  |-  F/ z  y  e.  B
63, 5nfan 1771 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
7 nfmpt2.3 . . . . 5  |-  F/_ z C
87nfeq2 2430 . . . 4  |-  F/ z  w  =  C
96, 8nfan 1771 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C )
109nfoprab 5900 . 2  |-  F/_ z { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
111, 10nfcxfr 2416 1  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   F/_wnfc 2406   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  nfof  6083  nfseq  11056  ptbasfi  17276  sdclem1  25865  fmuldfeqlem1  27124  stoweidlem51  27212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-oprab 5862  df-mpt2 5863
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