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Theorem dmmptg 5044
 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 eqid 2140 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21dmmpt 5042 . 2 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
3 elex 2700 . . . 4 (𝐵𝑉𝐵 ∈ V)
43ralimi 2498 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
5 rabid2 2610 . . 3 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
64, 5sylibr 133 . 2 (∀𝑥𝐴 𝐵𝑉𝐴 = {𝑥𝐴𝐵 ∈ V})
72, 6eqtr4id 2192 1 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ∀wral 2417  {crab 2421  Vcvv 2689   ↦ cmpt 3997  dom cdm 4547 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-mpt 3999  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560 This theorem is referenced by:  resfunexg  5649  rdgtfr  6279  rdgruledefgg  6280  negfi  11031  limccnp2lem  12853  dvmptclx  12888  dvmptaddx  12889  dvmptmulx  12890
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