Theorem imacosupp 7222

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 5230	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
2	1	imaeq1i 5382	. . . . . . 7 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
3		imaco 5557	. . . . . . 7 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2632	. . . . . 6 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
5	4	imaeq2i 5383	. . . . 5 ⊢ (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
6		funforn 6035	. . . . . . . 8 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺)
7	6	biimpi 205	. . . . . . 7 ⊢ (Fun 𝐺 → 𝐺:dom 𝐺–onto→ran 𝐺)
8	7	ad2antrl 760	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺–onto→ran 𝐺)
9		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ 𝑉)
10	9	anim2i 591	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ 𝐹 ∈ 𝑉))
11	10	ancomd 466	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V))
12		suppimacnv 7193	. . . . . . . . . . 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 3595	. . . . . . . . 9 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
15	14	biimpd 218	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
16	15	adantld 482	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
17	16	imp 444	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18		foimacnv 6067	. . . . . 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 691	. . . . 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 2656	. . . 4 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
21		coexg 7010	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
22	21	anim2i 591	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ (𝐹 ∘ 𝐺) ∈ V))
23	22	ancomd 466	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
24		suppimacnv 7193	. . . . . . 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 5385	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))))
27	26	adantr 480	. . . 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 480	. . . 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 2654	. . 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 628	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31		ima0 5400	. . . 4 ⊢ (𝐺 “ ∅) = ∅
32		id 22	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
33	32	intnand 953	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
34		supp0prc 7185	. . . . . 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 5385	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ ∅))
37	32	intnand 953	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38		supp0prc 7185	. . . . 5 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
39	37, 38	syl 17	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
40	31, 36, 39	3eqtr4a 2670	. . 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 175	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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  –onto→wfo 5802  (class class class)co 6549   supp csupp 7182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183

This theorem is referenced by:  gsumval3lem1  18129  gsumval3lem2  18130