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Theorem elmpt2cl1 6281
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 6280 . 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 1652    e. wcel 1725  (class class class)co 6073    e. cmpt2 6075
This theorem is referenced by:  iccssico2  10974  mhmrcl1  14731  rhmrcl1  15812  cncfrss  18911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-dm 4880  df-iota 5410  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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