Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmmptdf Structured version   Visualization version   GIF version

Theorem dmmptdf 42301
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 6072 . 2 dom 𝐴 = {𝑥𝐵𝐶 ∈ V}
3 dmmptdf.x . . . 4 𝑥𝜑
4 dmmptdf.c . . . . 5 ((𝜑𝑥𝐵) → 𝐶𝑉)
54elexd 3418 . . . 4 ((𝜑𝑥𝐵) → 𝐶 ∈ V)
63, 5ralrimia 3396 . . 3 (𝜑 → ∀𝑥𝐵 𝐶 ∈ V)
7 rabid2 3284 . . 3 (𝐵 = {𝑥𝐵𝐶 ∈ V} ↔ ∀𝑥𝐵 𝐶 ∈ V)
86, 7sylibr 237 . 2 (𝜑𝐵 = {𝑥𝐵𝐶 ∈ V})
92, 8eqtr4id 2792 1 (𝜑 → dom 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wnf 1790  wcel 2114  wral 3053  {crab 3057  Vcvv 3398  cmpt 5110  dom cdm 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538
This theorem is referenced by:  smfpimltmpt  43821  smfpimltxrmpt  43833  smfadd  43839  smfpimgtmpt  43855  smfpimgtxrmpt  43858  smfpimioompt  43859  smfrec  43862  smfmul  43868  smfmulc1  43869  smffmpt  43877  smfsupmpt  43887  smfinfmpt  43891  smflimsupmpt  43901  smfliminfmpt  43904
  Copyright terms: Public domain W3C validator