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Theorem dmmptg 4818
 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 2566 . . . 4 (𝐵𝑉𝐵 ∈ V)
21ralimi 2384 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 rabid2 2486 . . 3 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
42, 3sylibr 137 . 2 (∀𝑥𝐴 𝐵𝑉𝐴 = {𝑥𝐴𝐵 ∈ V})
5 eqid 2040 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65dmmpt 4816 . 2 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
74, 6syl6reqr 2091 1 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557   ↦ cmpt 3818  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  resfunexg  5382  rdgtfr  5961  rdgruledefgg  5962
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