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Theorem dmmptg 4882
Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
Assertion
Ref Expression
dmmptg (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem dmmptg
StepHypRef Expression
1 elex 2621 . . . 4 (𝐵𝑉𝐵 ∈ V)
21ralimi 2432 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 rabid2 2536 . . 3 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
42, 3sylibr 132 . 2 (∀𝑥𝐴 𝐵𝑉𝐴 = {𝑥𝐴𝐵 ∈ V})
5 eqid 2083 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65dmmpt 4880 . 2 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
74, 6syl6reqr 2134 1 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  wral 2353  {crab 2357  Vcvv 2612  cmpt 3865  dom cdm 4401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-mpt 3867  df-xp 4407  df-rel 4408  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414
This theorem is referenced by:  resfunexg  5458  rdgtfr  6071  rdgruledefgg  6072  negfi  10484
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