Theorem imacosuppOLD 7875

Description: Obsolete version of imacosupp 7874 as of 15-Sep-2023. The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem imacosuppOLD

Step	Hyp	Ref	Expression
1		cnvco 5756	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
2	1	imaeq1i 5926	. . . . . . 7 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
3		imaco 6104	. . . . . . 7 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2844	. . . . . 6 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
5	4	imaeq2i 5927	. . . . 5 ⊢ (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
6		funforn 6597	. . . . . . . 8 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺)
7	6	biimpi 218	. . . . . . 7 ⊢ (Fun 𝐺 → 𝐺:dom 𝐺–onto→ran 𝐺)
8	7	ad2antrl 726	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺–onto→ran 𝐺)
9		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ 𝑉)
10	9	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ 𝐹 ∈ 𝑉))
11	10	ancomd 464	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V))
12		suppimacnv 7841	. . . . . . . . . . 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 3998	. . . . . . . . 9 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
15	14	biimpd 231	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
16	15	adantld 493	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
17	16	imp 409	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18		foimacnv 6632	. . . . . 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 586	. . . . 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 2868	. . . 4 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
21		coexg 7634	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
22	21	anim2i 618	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ (𝐹 ∘ 𝐺) ∈ V))
23	22	ancomd 464	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
24		suppimacnv 7841	. . . . . . 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 5929	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))))
27	26	adantr 483	. . . 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 483	. . . 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 2866	. . 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 422	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31		ima0 5945	. . . 4 ⊢ (𝐺 “ ∅) = ∅
32		id 22	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
33	32	intnand 491	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
34		supp0prc 7833	. . . . . 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 5929	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ ∅))
37	32	intnand 491	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38		supp0prc 7833	. . . . 5 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
39	37, 38	syl 17	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
40	31, 36, 39	3eqtr4a 2882	. . 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 184	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 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  {csn 4567  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   “ cima 5558   ∘ ccom 5559  Fun wfun 6349  –onto→wfo 6353  (class class class)co 7156   supp csupp 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fo 6361  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-supp 7831

This theorem is referenced by: (None)