Theorem elmpocl2 6201

Description: If a two-parameter class is inhabited, 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
elmpocl.f	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
elmpocl2	|- ( 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 elmpocl2

Step	Hyp	Ref	Expression
1		elmpocl.f	. . 3 |- F = ( x e. A , y e. B |-> C )
2	1	elmpocl 6199	. 2 |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) )
3	2	simprd 114	1 |- ( X e. ( S F T ) -> T e. B )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6000   e. cmpo 6002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  iccssico2  10139  elfzoel2  10338  mhmrcl2  13492  rhmrcl2  14114  cncfrss2  15244