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Theorem elmpt2cl 5726
Description: If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpt2cl.f  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
elmpt2cl  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
Distinct variable groups:    x, A, y   
x, B, y
Allowed substitution hints:    C( x, y)    S( x, y)    T( x, y)    F( x, y)    X( x, y)

Proof of Theorem elmpt2cl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elmpt2cl.f . . . . . 6  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 df-mpt2 5545 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2076 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43dmeqi 4564 . . . 4  |-  dom  F  =  dom  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 dmoprabss 5614 . . . 4  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  C_  ( A  X.  B
)
64, 5eqsstri 3003 . . 3  |-  dom  F  C_  ( A  X.  B
)
71mpt2fun 5631 . . . . . 6  |-  Fun  F
8 funrel 4947 . . . . . 6  |-  ( Fun 
F  ->  Rel  F )
97, 8ax-mp 7 . . . . 5  |-  Rel  F
10 relelfvdm 5233 . . . . 5  |-  ( ( Rel  F  /\  X  e.  ( F `  <. S ,  T >. )
)  ->  <. S ,  T >.  e.  dom  F
)
119, 10mpan 408 . . . 4  |-  ( X  e.  ( F `  <. S ,  T >. )  ->  <. S ,  T >.  e.  dom  F )
12 df-ov 5543 . . . 4  |-  ( S F T )  =  ( F `  <. S ,  T >. )
1311, 12eleq2s 2148 . . 3  |-  ( X  e.  ( S F T )  ->  <. S ,  T >.  e.  dom  F
)
146, 13sseldi 2971 . 2  |-  ( X  e.  ( S F T )  ->  <. S ,  T >.  e.  ( A  X.  B ) )
15 opelxp 4402 . 2  |-  ( <. S ,  T >.  e.  ( A  X.  B
)  <->  ( S  e.  A  /\  T  e.  B ) )
1614, 15sylib 131 1  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   <.cop 3406    X. cxp 4371   dom cdm 4373   Rel wrel 4378   Fun wfun 4924   ` cfv 4930  (class class class)co 5540   {coprab 5541    |-> cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  elmpt2cl1  5727  elmpt2cl2  5728  elovmpt2  5729  ixxssxr  8870  elixx3g  8871  ixxssixx  8872  eliooxr  8897  elfz2  8983
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