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Theorem dmmpog 6119
 Description: Domain of an operation given by the maps-to notation, closed form of dmmpo 6115. 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 108 . . 3 ((𝐶𝑉 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑉)
21ralrimivva 2519 . 2 (𝐶𝑉 → ∀𝑥𝐴𝑦𝐵 𝐶𝑉)
3 dmmpog.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43dmmpoga 6118 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
52, 4syl 14 1 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2112  ∀wral 2418   × cxp 4549  dom cdm 4551   ∈ cmpo 5788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-un 4366 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-fv 5143  df-oprab 5790  df-mpo 5791  df-1st 6050  df-2nd 6051 This theorem is referenced by: (None)
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