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Theorem imacosupp 7872
 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 7868 . . . 4 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
21imaeq2d 5899 . . 3 ((𝐹𝑉𝐺𝑊) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))))
3 funforn 6577 . . . 4 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
4 foimacnv 6613 . . . 4 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
53, 4sylanb 584 . . 3 ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
62, 5sylan9eq 2853 . 2 (((𝐹𝑉𝐺𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
76ex 416 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 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3882  ◡ccnv 5521  dom cdm 5522  ran crn 5523   “ cima 5525   ∘ ccom 5526  Fun wfun 6323  –onto→wfo 6327  (class class class)co 7142   supp csupp 7823 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fo 6335  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-supp 7824 This theorem is referenced by:  gsumval3lem1  19036  gsumval3lem2  19037
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