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Theorem elovmpt2 6064
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 14683, islmhm 15784. (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 6061 . . 3  |-  ( F  e.  ( X D Y )  ->  ( X  e.  A  /\  Y  e.  B )
)
3 elovmpt2.c . . . . . . 7  |-  C  e. 
_V
43gen2 1534 . . . . . 6  |-  A. a A. b  C  e.  _V
5 elovmpt2.e . . . . . . . 8  |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )
65eleq1d 2349 . . . . . . 7  |-  ( ( a  =  X  /\  b  =  Y )  ->  ( C  e.  _V  <->  E  e.  _V ) )
76spc2gv 2871 . . . . . 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 5977 . . . . 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 2350 . . 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 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  isgim  14726  oppglsm  14953  islmim  15815
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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