Theorem elmpt2cl1 5719

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

Step	Hyp	Ref	Expression
1		elmpt2cl.f	. . 3 |- F = ( x e. A , y e. B |-> C )
2	1	elmpt2cl 5718	. 2 |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) )
3	2	simpld 110	1 |- ( X e. ( S F T ) -> S e. A )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433  (class class class)co 5532   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  iccssico2  8970  elfzoel1  9155