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

Theorem imacosupp 8234
Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
Assertion
Ref Expression
imacosupp ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Proof of Theorem imacosupp
StepHypRef Expression
1 suppco 8231 . . . 4 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
21imaeq2d 6078 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))))
3 funforn 6827 . . . 4 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
4 foimacnv 6865 . . . 4 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
53, 4sylanb 581 . . 3 ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
62, 5sylan9eq 2797 . 2 (((𝐹𝑉𝐺𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
76ex 412 1 ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555  ontowfo 6559  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  gsumval3lem1  19923  gsumval3lem2  19924
  Copyright terms: Public domain W3C validator