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Theorem elmpocl2 7369
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
elmpocl.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpocl2 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇𝐵)
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
StepHypRef Expression
1 elmpocl.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21elmpocl 7367 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
32simprd 499 1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  (class class class)co 7135  cmpo 7137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140
This theorem is referenced by:  iccssico2  12799  swrdcl  13998  pfxcl  14030  mhmrcl2  17952  rhmrcl2  19468  mpfrcl  20757  cncfrss2  23497  relowlpssretop  34781
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