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

Theorem dmmptssf 38948
Description: The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
dmmptssf.1 𝑥𝐴
dmmptssf.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmptssf dom 𝐹𝐴

Proof of Theorem dmmptssf
StepHypRef Expression
1 dmmptssf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
21dmmpt 5599 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 dmmptssf.1 . . 3 𝑥𝐴
43ssrab2f 38824 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
52, 4eqsstri 3620 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wnfc 2748  {crab 2912  Vcvv 3190  wss 3560  cmpt 4683  dom cdm 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-mpt 4685  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by:  limsupequzmpt2  39386
  Copyright terms: Public domain W3C validator