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

Theorem dmmpog 8098
Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 8095. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
Hypothesis
Ref Expression
dmmpog.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpog (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem dmmpog
StepHypRef Expression
1 simpl 482 . . 3 ((𝐶𝑉 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑉)
21ralrimivva 3200 . 2 (𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶𝑉)
3 dmmpog.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43dmmpoga 8097 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
52, 4syl 17 1 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   × cxp 5687  dom cdm 5689  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014
This theorem is referenced by:  aovmpt4g  47151
  Copyright terms: Public domain W3C validator