Theorem imacosupp 7335

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		cnvco 5308	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
2	1	imaeq1i 5463	. . . . . . 7 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
3		imaco 5640	. . . . . . 7 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2644	. . . . . 6 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
5	4	imaeq2i 5464	. . . . 5 ⊢ (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
6		funforn 6122	. . . . . . . 8 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺)
7	6	biimpi 206	. . . . . . 7 ⊢ (Fun 𝐺 → 𝐺:dom 𝐺–onto→ran 𝐺)
8	7	ad2antrl 764	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺–onto→ran 𝐺)
9		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ 𝑉)
10	9	anim2i 593	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ 𝐹 ∈ 𝑉))
11	10	ancomd 467	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V))
12		suppimacnv 7306	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
14	13	sseq1d 3632	. . . . . . . . 9 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
15	14	biimpd 219	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
16	15	adantld 483	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
17	16	imp 445	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18		foimacnv 6154	. . . . . 6 ⊢ ((𝐺:dom 𝐺–onto→ran 𝐺 ∧ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))) = (◡𝐹 “ (V ∖ {𝑍})))
19	8, 17, 18	syl2anc 693	. . . . 5 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))) = (◡𝐹 “ (V ∖ {𝑍})))
20	5, 19	syl5eq 2668	. . . 4 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
21		coexg 7117	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
22	21	anim2i 593	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ (𝐹 ∘ 𝐺) ∈ V))
23	22	ancomd 467	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
24		suppimacnv 7306	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
25	23, 24	syl 17	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
26	25	imaeq2d 5466	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))))
27	26	adantr 481	. . . 4 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))))
28	13	adantr 481	. . . 4 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
29	20, 27, 28	3eqtr4d 2666	. . 3 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
30	29	exp31 630	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31		ima0 5481	. . . 4 ⊢ (𝐺 “ ∅) = ∅
32		id 22	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
33	32	intnand 962	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
34		supp0prc 7298	. . . . . 6 ⊢ (¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
35	33, 34	syl 17	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
36	35	imaeq2d 5466	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ ∅))
37	32	intnand 962	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38		supp0prc 7298	. . . . 5 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
39	37, 38	syl 17	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
40	31, 36, 39	3eqtr4a 2682	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
41	40	2a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
42	30, 41	pm2.61i 176	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  Fun wfun 5882  –onto→wfo 5886  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  gsumval3lem1  18306  gsumval3lem2  18307