Theorem imacosupp 8149

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	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Proof of Theorem imacosupp

Step	Hyp	Ref	Expression
1		suppco 8146	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
2	1	imaeq2d 6012	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))))
3		funforn 6746	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺)
4		foimacnv 6784	. . . 4 ⊢ ((𝐺:dom 𝐺–onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
5	3, 4	sylanb 587	. . 3 ⊢ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
6	2, 5	sylan9eq 2794	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
7	6	ex 413	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   “ cima 5621   ∘ ccom 5622  Fun wfun 6479  –onto→wfo 6483  (class class class)co 7356   supp csupp 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-supp 8101

This theorem is referenced by:  gsumval3lem1  19871  gsumval3lem2  19872