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

Theorem dmfco 6994
Description: Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
Assertion
Ref Expression
dmfco ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))

Proof of Theorem dmfco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldm2g 5902 . . . 4 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺)))
2 opelco2g 5870 . . . . . 6 ((𝐴 ∈ dom 𝐺𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
32elvd 3478 . . . . 5 (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
43exbidv 1917 . . . 4 (𝐴 ∈ dom 𝐺 → (∃𝑦𝐴, 𝑦⟩ ∈ (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
51, 4bitrd 279 . . 3 (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
65adantl 481 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7 fvex 6910 . . . 4 (𝐺𝐴) ∈ V
87eldm2 5904 . . 3 ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
9 opeq1 4874 . . . . . . 7 (𝑥 = (𝐺𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺𝐴), 𝑦⟩)
109eleq1d 2814 . . . . . 6 (𝑥 = (𝐺𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹))
117, 10ceqsexv 3523 . . . . 5 (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹)
12 eqcom 2735 . . . . . . . 8 (𝑥 = (𝐺𝐴) ↔ (𝐺𝐴) = 𝑥)
13 funopfvb 6953 . . . . . . . 8 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1412, 13bitrid 283 . . . . . . 7 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
1514anbi1d 630 . . . . . 6 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1615exbidv 1917 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1711, 16bitr3id 285 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
1817exbidv 1917 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
198, 18bitrid 283 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) ∈ dom 𝐹 ↔ ∃𝑦𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
206, 19bitr4d 282 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  Vcvv 3471  cop 4635  dom cdm 5678  ccom 5682  Fun wfun 6542  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556
This theorem is referenced by:  hoicvr  45936  funressnfv  46425  dmfcoafv  46555  afvco2  46556
  Copyright terms: Public domain W3C validator