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Theorem dmmptdf 41355
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.x . . . 4 𝑥𝜑
2 dmmptdf.c . . . . 5 ((𝜑𝑥𝐵) → 𝐶𝑉)
32elexd 3519 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
41, 3ralrimia 41265 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
5 rabid2 3386 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
64, 5sylibr 235 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
7 dmmptdf.a . . 3 𝐴 = (𝑥𝐵𝐶)
87dmmpt 6091 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
96, 8syl6reqr 2879 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wnf 1777  wcel 2107  wral 3142  {crab 3146  Vcvv 3499  cmpt 5142  dom cdm 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566
This theorem is referenced by:  smfpimltmpt  42891  smfpimltxrmpt  42903  smfadd  42909  smfpimgtmpt  42925  smfpimgtxrmpt  42928  smfpimioompt  42929  smfrec  42932  smfmul  42938  smfmulc1  42939  smffmpt  42947  smfsupmpt  42957  smfinfmpt  42961  smflimsupmpt  42971  smfliminfmpt  42974
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