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Theorem dmmpog 8055
Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 8051. 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 3192 . 2 (𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶𝑉)
3 dmmpog.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43dmmpoga 8053 . 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 1533  wcel 2098  wral 3053   × cxp 5665  dom cdm 5667  cmpo 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970
This theorem is referenced by:  aovmpt4g  46455
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