MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmfex Structured version   Visualization version   GIF version

Theorem dmfex 7272
Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dmfex ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)

Proof of Theorem dmfex
StepHypRef Expression
1 fdm 6192 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 dmexg 7245 . . . 4 (𝐹𝐶 → dom 𝐹 ∈ V)
3 eleq1 2838 . . . 4 (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
42, 3syl5ib 234 . . 3 (dom 𝐹 = 𝐴 → (𝐹𝐶𝐴 ∈ V))
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹𝐶𝐴 ∈ V))
65impcom 394 1 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  dom cdm 5250  wf 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261  df-fn 6035  df-f 6036
This theorem is referenced by:  wemoiso  7301  fopwdom  8225  fowdom  8633  wdomfil  9085  fin23lem17  9363  fin23lem32  9369  fin23lem39  9375  enfin1ai  9409  fin1a2lem7  9431  lindfmm  20384  kelac1  38160
  Copyright terms: Public domain W3C validator