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Theorem dmmptg 5121
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 2177 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
21dmmpt 5119 . 2 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
3 elex 2748 . . . 4 (𝐵𝑉𝐵 ∈ V)
43ralimi 2540 . . 3 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
5 rabid2 2653 . . 3 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
64, 5sylibr 134 . 2 (∀𝑥𝐴 𝐵𝑉𝐴 = {𝑥𝐴𝐵 ∈ V})
72, 6eqtr4id 2229 1 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  {crab 2459  Vcvv 2737  cmpt 4061  dom cdm 4622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-mpt 4063  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635
This theorem is referenced by:  resfunexg  5732  rdgtfr  6368  rdgruledefgg  6369  negfi  11207  limccnp2lem  13778  dvmptclx  13813  dvmptaddx  13814  dvmptmulx  13815
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