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

Theorem dmmpti 4971
 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 = (x AB)
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 = (x AB)
31, 2fnmpti 4970 . 2 𝐹 Fn A
4 fndm 4941 . 2 (𝐹 Fn A → dom 𝐹 = A)
53, 4ax-mp 7 1 dom 𝐹 = A
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ↦ cmpt 3809  dom cdm 4288   Fn wfn 4840 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848 This theorem is referenced by:  brtpos2  5807
 Copyright terms: Public domain W3C validator