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Theorem elmpt2cl1 6062
Description: If a two-parameter class is not empty, 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
elmpt2cl.f  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
elmpt2cl1  |-  ( 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 elmpt2cl1
StepHypRef Expression
1 elmpt2cl.f . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21elmpt2cl 6061 . 2  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
32simpld 445 1  |-  ( X  e.  ( S F T )  ->  S  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  iccssico2  10723  mhmrcl1  14418  rhmrcl1  15499  cncfrss  18395
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-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-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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