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 6015	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))))
3		funforn 6747	. . . 4 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺)
4		foimacnv 6785	. . . 4 ⊢ ((𝐺:dom 𝐺–onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
5	3, 4	sylanb 581	. . 3 ⊢ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
6	2, 5	sylan9eq 2784	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
7	6	ex 412	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 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3905  ^◡ccnv 5622  dom cdm 5623  ran crn 5624   “ cima 5626   ∘ ccom 5627  Fun wfun 6480  –onto→wfo 6484  (class class class)co 7353   supp csupp 8100

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-supp 8101

This theorem is referenced by:  gsumval3lem1  19803  gsumval3lem2  19804