ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfmpo Unicode version

Theorem nfmpo 5940
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 5876 . 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 2313 . . . . 5  |-  F/ z  x  e.  A
4 nfmpo.2 . . . . . 6  |-  F/_ z B
54nfcri 2313 . . . . 5  |-  F/ z  y  e.  B
63, 5nfan 1565 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
7 nfmpo.3 . . . . 5  |-  F/_ z C
87nfeq2 2331 . . . 4  |-  F/ z  w  =  C
96, 8nfan 1565 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C )
109nfoprab 5923 . 2  |-  F/_ z { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) }
111, 10nfcxfr 2316 1  |-  F/_ z
( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   F/_wnfc 2306   {coprab 5872    e. cmpo 5873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-oprab 5875  df-mpo 5876
This theorem is referenced by:  nfof  6084  nfseq  10449
  Copyright terms: Public domain W3C validator