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Theorem elmpt2cl2 6503
Description: If a two-parameter class is not empty, the second 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
elmpt2cl2  |-  ( X  e.  ( S F T )  ->  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 elmpt2cl2
StepHypRef Expression
1 elmpt2cl.f . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21elmpt2cl 6501 . 2  |-  ( X  e.  ( S F T )  ->  ( S  e.  A  /\  T  e.  B )
)
32simprd 463 1  |-  ( X  e.  ( S F T )  ->  T  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767  (class class class)co 6284    |-> cmpt2 6286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5551  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289
This theorem is referenced by:  iccssico2  11598  swrdcl  12609  mhmrcl2  15790  rhmrcl2  17170  mpfrcl  17986  cncfrss2  21159
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