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

Theorem imacosupp 8176
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 8173 . . . 4 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
21imaeq2d 6049 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))))
3 funforn 6799 . . . 4 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
4 foimacnv 6837 . . . 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 2791 . 2 (((𝐹𝑉𝐺𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
76ex 413 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 396   = wceq 1541  wcel 2106  wss 3944  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  ccom 5673  Fun wfun 6526  ontowfo 6530  (class class class)co 7393   supp csupp 8128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-supp 8129
This theorem is referenced by:  gsumval3lem1  19732  gsumval3lem2  19733
  Copyright terms: Public domain W3C validator