Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elmpt2cl2 Structured version   Visualization version   GIF version

Theorem elmpt2cl2 7035
 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 7033 . 2 (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆𝐴𝑇𝐵))
32simprd 482 1 (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1624   ∈ wcel 2131  (class class class)co 6805   ↦ cmpt2 6807 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-xp 5264  df-dm 5268  df-iota 6004  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810 This theorem is referenced by:  iccssico2  12432  swrdcl  13610  mhmrcl2  17532  rhmrcl2  18914  mpfrcl  19712  cncfrss2  22888  relowlpssretop  33515  pfxcl  41888
 Copyright terms: Public domain W3C validator