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Theorem dmmptdf 42763
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
dmmptdf.x 𝑥𝜑
dmmptdf.a 𝐴 = (𝑥𝐵𝐶)
dmmptdf.c ((𝜑𝑥𝐵) → 𝐶𝑉)
Assertion
Ref Expression
dmmptdf (𝜑 → dom 𝐴 = 𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem dmmptdf
StepHypRef Expression
1 dmmptdf.a . . 3 𝐴 = (𝑥𝐵𝐶)
21dmmpt 6143 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
3 dmmptdf.x . . . 4 𝑥𝜑
4 dmmptdf.c . . . . 5 ((𝜑𝑥𝐵) → 𝐶𝑉)
54elexd 3452 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
63, 5ralrimia 3430 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 rabid2 3314 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
86, 7sylibr 233 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
92, 8eqtr4id 2797 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wnf 1786  wcel 2106  wral 3064  {crab 3068  Vcvv 3432  cmpt 5157  dom cdm 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602
This theorem is referenced by:  smfpimltmpt  44282  smfpimltxrmpt  44294  smfadd  44300  smfpimgtmpt  44316  smfpimgtxrmpt  44319  smfpimioompt  44320  smfrec  44323  smfmul  44329  smfmulc1  44330  smffmpt  44338  smfsupmpt  44348  smfinfmpt  44352  smflimsupmpt  44362  smfliminfmpt  44365
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