MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpt2g Structured version   Visualization version   GIF version

Theorem dmmpt2g 7240
Description: Domain of an operation given by the "maps to" notation, closed form of dmmpt2 7237. 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.) (Prove shortened by AV, 10-Feb-2019.)
Hypothesis
Ref Expression
dmmpt2g.f 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
dmmpt2g (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 dmmpt2g
StepHypRef Expression
1 simpl 473 . . 3 ((𝐶𝑉 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑉)
21ralrimivva 2970 . 2 (𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶𝑉)
3 dmmpt2g.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43dmmpt2ga 7239 . 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 384   = wceq 1482  wcel 1989  wral 2911   × cxp 5110  dom cdm 5112  cmpt2 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166
This theorem is referenced by:  aovmpt4g  41050
  Copyright terms: Public domain W3C validator