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Theorem elmpt2cl2 7112
 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 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elmpt2cl2 (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 elmpt2cl2
StepHypRef Expression
1 elmpt2cl.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21elmpt2cl 7110 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
32simprd 490 1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  (class class class)co 6878   ↦ cmpt2 6880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-dm 5322  df-iota 6064  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883 This theorem is referenced by:  iccssico2  12496  swrdcl  13669  pfxcl  13720  mhmrcl2  17654  rhmrcl2  19038  mpfrcl  19840  cncfrss2  23023  relowlpssretop  33710
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