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Theorem elovmpt2 6505
Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 16081, islmhm 17485. (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 6502 . . 3  |-  ( F  e.  ( X D Y )  ->  ( X  e.  A  /\  Y  e.  B )
)
3 elovmpt2.c . . . . . . 7  |-  C  e. 
_V
43gen2 1602 . . . . . 6  |-  A. a A. b  C  e.  _V
5 elovmpt2.e . . . . . . . 8  |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )
65eleq1d 2536 . . . . . . 7  |-  ( ( a  =  X  /\  b  =  Y )  ->  ( C  e.  _V  <->  E  e.  _V ) )
76spc2gv 3201 . . . . . 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 6417 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B  /\  E  e.  _V )  ->  ( X D Y )  =  E )
108, 9mpd3an3 1325 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X D Y )  =  E )
1110eleq2d 2537 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( F  e.  ( X D Y )  <-> 
F  e.  E ) )
122, 11biadan2 642 . 2  |-  ( F  e.  ( X D Y )  <->  ( ( X  e.  A  /\  Y  e.  B )  /\  F  e.  E
) )
13 df-3an 975 . 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 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   _Vcvv 3113  (class class class)co 6285    |-> cmpt2 6287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290
This theorem is referenced by:  isgim  16124  oppglsm  16477  islmim  17520  wlkcompim  24299  wlkelwrd  24303  clwlkcompim  24537
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