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Theorem elovmpt2 6106
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 14732, islmhm 15833. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses
Ref Expression
elovmpt2.d  |-  D  =  ( a  e.  A ,  b  e.  B  |->  C )
elovmpt2.c  |-  C  e. 
_V
elovmpt2.e  |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )
Assertion
Ref Expression
elovmpt2  |-  ( F  e.  ( X D Y )  <->  ( X  e.  A  /\  Y  e.  B  /\  F  e.  E ) )
Distinct variable groups:    A, a,
b    B, a, b    E, a, b    F, a, b    X, a, b    Y, a, b
Allowed substitution hints:    C( a, b)    D( a, b)

Proof of Theorem elovmpt2
StepHypRef Expression
1 elovmpt2.d . . . 4  |-  D  =  ( a  e.  A ,  b  e.  B  |->  C )
21elmpt2cl 6103 . . 3  |-  ( F  e.  ( X D Y )  ->  ( X  e.  A  /\  Y  e.  B )
)
3 elovmpt2.c . . . . . . 7  |-  C  e. 
_V
43gen2 1538 . . . . . 6  |-  A. a A. b  C  e.  _V
5 elovmpt2.e . . . . . . . 8  |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )
65eleq1d 2382 . . . . . . 7  |-  ( ( a  =  X  /\  b  =  Y )  ->  ( C  e.  _V  <->  E  e.  _V ) )
76spc2gv 2905 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( A. a A. b  C  e.  _V  ->  E  e.  _V )
)
84, 7mpi 16 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  E  e.  _V )
95, 1ovmpt2ga 6019 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B  /\  E  e.  _V )  ->  ( X D Y )  =  E )
108, 9mpd3an3 1278 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X D Y )  =  E )
1110eleq2d 2383 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( F  e.  ( X D Y )  <-> 
F  e.  E ) )
122, 11biadan2 623 . 2  |-  ( F  e.  ( X D Y )  <->  ( ( X  e.  A  /\  Y  e.  B )  /\  F  e.  E
) )
13 df-3an 936 . 2  |-  ( ( X  e.  A  /\  Y  e.  B  /\  F  e.  E )  <->  ( ( X  e.  A  /\  Y  e.  B
)  /\  F  e.  E ) )
1412, 13bitr4i 243 1  |-  ( F  e.  ( X D Y )  <->  ( X  e.  A  /\  Y  e.  B  /\  F  e.  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1531    = wceq 1633    e. wcel 1701   _Vcvv 2822  (class class class)co 5900    e. cmpt2 5902
This theorem is referenced by:  isgim  14775  oppglsm  15002  islmim  15864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905
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