Theorem suppco 8146

Description: The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 8148. (Revised by SN, 15-Sep-2023.)

Assertion

Ref	Expression
suppco	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

Proof of Theorem suppco

Step	Hyp	Ref	Expression
1		coexg 7869	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
2		simpl 483	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝑍 ∈ V)
3		suppimacnv 8114	. . . . 5 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
4	1, 2, 3	syl2an2 692	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
5		cnvco 5827	. . . . . 6 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
6	5	imaeq1i 6009	. . . . 5 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
7	6	a1i 11	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})))
8		imaco 6202	. . . . 5 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
9		simprl 776	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝐹 ∈ 𝑉)
10		suppimacnv 8114	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
11	9, 2, 10	syl2anc 590	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	11	imaeq2d 6012	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
13	8, 12	eqtr4id 2793	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
14	4, 7, 13	3eqtrd 2778	. . 3 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
15	14	ex 413	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))))
16		prcnel 3456	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
17	16	intnand 489	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
18		supp0prc 8103	. . . . 5 ⊢ (¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
19	17, 18	syl 17	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
20	16	intnand 489	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21		supp0prc 8103	. . . . . . 7 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
22	20, 21	syl 17	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
23	22	imaeq2d 6012	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
24		ima0 6029	. . . . 5 ⊢ (◡𝐺 “ ∅) = ∅
25	23, 24	eqtrdi 2790	. . . 4 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
26	19, 25	eqtr4d 2777	. . 3 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
27	26	a1d 25	. 2 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))))
28	15, 27	pm2.61i 183	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880  ∅c0 4261  {csn 4555  ^◡ccnv 5617   “ cima 5621   ∘ ccom 5622  (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-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-supp 8101

This theorem is referenced by:  supp0cosupp0  8148  imacosupp  8149  extvfvcl  33720