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Theorem elmpt2cl1 6230
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 6229 . 2  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
32simpld 446 1  |-  ( X  e.  ( S F T )  ->  S  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717  (class class class)co 6022    e. cmpt2 6024
This theorem is referenced by:  iccssico2  10918  mhmrcl1  14670  rhmrcl1  15751  cncfrss  18794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-xp 4826  df-dm 4830  df-iota 5360  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027
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