Theorem suppco 8204

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 8206. (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 7928	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
2		simpl 488	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝑍 ∈ V)
3		suppimacnv 8172	. . . . 5 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
4	1, 2, 3	syl2an2 699	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
5		cnvco 5877	. . . . . 6 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
6	5	imaeq1i 6061	. . . . 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 6254	. . . . 5 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
9		simprl 783	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝐹 ∈ 𝑉)
10		suppimacnv 8172	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
11	9, 2, 10	syl2anc 596	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	11	imaeq2d 6064	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
13	8, 12	eqtr4id 2819	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
14	4, 7, 13	3eqtrd 2804	. . 3 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
15	14	ex 418	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))))
16		prcnel 3482	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
17	16	intnand 494	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
18		supp0prc 8161	. . . . 5 ⊢ (¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
19	17, 18	syl 18	. . . 4 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
20	16	intnand 494	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21		supp0prc 8161	. . . . . . 7 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
22	20, 21	syl 18	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
23	22	imaeq2d 6064	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
24		ima0 6081	. . . . 5 ⊢ (◡𝐺 “ ∅) = ∅
25	23, 24	eqtrdi 2816	. . . 4 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
26	19, 25	eqtr4d 2803	. . 3 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
27	26	a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))))
28	15, 27	pm2.61i 184	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∖ cdif 3903  ∅c0 4286  {csn 4591  ^◡ccnv 5662   “ cima 5666   ∘ ccom 5667  (class class class)co 7416   supp csupp 8158

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-supp 8159

This theorem is used by:  supp0cosupp0  8206  imacosupp  8207  extvfvcl  33966