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Theorem elmpt2cl1 5843
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 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpt2cl1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆𝐴)
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 = (𝑥𝐴, 𝑦𝐵𝐶)
21elmpt2cl 5842 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
32simpld 110 1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  (class class class)co 5652  cmpt2 5654
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657
This theorem is referenced by:  pmresg  6431  iccssico2  9363  elfzoel1  9552  cncfrss  11586
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