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Theorem elmpocl1 6032
Description: If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
elmpocl.f  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
elmpocl1  |-  ( X  e.  ( S F T )  ->  S  e.  A )
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 elmpocl1
StepHypRef Expression
1 elmpocl.f . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21elmpocl 6031 . 2  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
32simpld 111 1  |-  ( X  e.  ( S F T )  ->  S  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1342    e. wcel 2135  (class class class)co 5837    e. cmpo 5839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-iota 5148  df-fun 5185  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842
This theorem is referenced by:  pmresg  6634  iccssico2  9875  elfzoel1  10071  cncfrss  13129  limccl  13195
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