Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmfcoafv Structured version   Visualization version   GIF version

Theorem dmfcoafv 40556
Description: Domains of a function composition, analogous to dmfco 6229. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
dmfcoafv ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Proof of Theorem dmfcoafv
StepHypRef Expression
1 dmfco 6229 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
2 funres 5887 . . . . . . 7 (Fun 𝐺 → Fun (𝐺 ↾ {𝐴}))
32anim2i 592 . . . . . 6 ((𝐴 ∈ dom 𝐺 ∧ Fun 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
43ancoms 469 . . . . 5 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
5 df-dfat 40497 . . . . . 6 (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
6 afvfundmfveq 40519 . . . . . 6 (𝐺 defAt 𝐴 → (𝐺'''𝐴) = (𝐺𝐴))
75, 6sylbir 225 . . . . 5 ((𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})) → (𝐺'''𝐴) = (𝐺𝐴))
84, 7syl 17 . . . 4 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐺'''𝐴) = (𝐺𝐴))
98eqcomd 2627 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐺𝐴) = (𝐺'''𝐴))
109eleq1d 2683 . 2 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐺𝐴) ∈ dom 𝐹 ↔ (𝐺'''𝐴) ∈ dom 𝐹))
111, 10bitrd 268 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4148  dom cdm 5074  cres 5076  ccom 5078  Fun wfun 5841  cfv 5847   defAt wdfat 40494  '''cafv 40495
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 4741  ax-nul 4749  ax-pr 4867
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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-dfat 40497  df-afv 40498
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator