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Theorem nfmpt2 5573
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 5517 . 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 2172 . . . . 5  |-  F/ z  x  e.  A
4 nfmpt2.2 . . . . . 6  |-  F/_ z B
54nfcri 2172 . . . . 5  |-  F/ z  y  e.  B
63, 5nfan 1457 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
7 nfmpt2.3 . . . . 5  |-  F/_ z C
87nfeq2 2189 . . . 4  |-  F/ z  w  =  C
96, 8nfan 1457 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C )
109nfoprab 5557 . 2  |-  F/_ z { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
111, 10nfcxfr 2175 1  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243    e. wcel 1393   F/_wnfc 2165   {coprab 5513    |-> cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-oprab 5516  df-mpt2 5517
This theorem is referenced by:  nfiseq  9218
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