ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmmpti GIF version

Theorem dmmpti 5302
Description: Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fnmpti.1 𝐵 ∈ V
fnmpti.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmpti dom 𝐹 = 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem dmmpti
StepHypRef Expression
1 fnmpti.1 . . 3 𝐵 ∈ V
2 fnmpti.2 . . 3 𝐹 = (𝑥𝐴𝐵)
31, 2fnmpti 5301 . 2 𝐹 Fn 𝐴
4 fndm 5272 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4ax-mp 5 1 dom 𝐹 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  Vcvv 2712  cmpt 4028  dom cdm 4589   Fn wfn 5168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-fun 5175  df-fn 5176
This theorem is referenced by:  brtpos2  6201  dmtopon  12517
  Copyright terms: Public domain W3C validator