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

Theorem dmmpog 8008
Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 8004. 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 484 . . 3 ((𝐶𝑉 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑉)
21ralrimivva 3198 . 2 (𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶𝑉)
3 dmmpog.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43dmmpoga 8006 . 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 397   = wceq 1542  wcel 2107  wral 3065   × cxp 5632  dom cdm 5634  cmpo 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923
This theorem is referenced by:  aovmpt4g  45440
  Copyright terms: Public domain W3C validator