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Theorem elovmpt2 6493
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 16466, islmhm 17868. (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 6490 . . 3  |-  ( F  e.  ( X D Y )  ->  ( X  e.  A  /\  Y  e.  B )
)
3 elovmpt2.c . . . . . . 7  |-  C  e. 
_V
43gen2 1624 . . . . . 6  |-  A. a A. b  C  e.  _V
5 elovmpt2.e . . . . . . . 8  |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )
65eleq1d 2523 . . . . . . 7  |-  ( ( a  =  X  /\  b  =  Y )  ->  ( C  e.  _V  <->  E  e.  _V ) )
76spc2gv 3194 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( A. a A. b  C  e.  _V  ->  E  e.  _V )
)
84, 7mpi 17 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  E  e.  _V )
95, 1ovmpt2ga 6405 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B  /\  E  e.  _V )  ->  ( X D Y )  =  E )
108, 9mpd3an3 1323 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X D Y )  =  E )
1110eleq2d 2524 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( F  e.  ( X D Y )  <-> 
F  e.  E ) )
122, 11biadan2 640 . 2  |-  ( F  e.  ( X D Y )  <->  ( ( X  e.  A  /\  Y  e.  B )  /\  F  e.  E
) )
13 df-3an 973 . 2  |-  ( ( X  e.  A  /\  Y  e.  B  /\  F  e.  E )  <->  ( ( X  e.  A  /\  Y  e.  B
)  /\  F  e.  E ) )
1412, 13bitr4i 252 1  |-  ( F  e.  ( X D Y )  <->  ( X  e.  A  /\  Y  e.  B  /\  F  e.  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398    e. wcel 1823   _Vcvv 3106  (class class class)co 6270    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  isgim  16509  oppglsm  16861  islmim  17903  wlkcompim  24728  wlkelwrd  24732  clwlkcompim  24966
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