Theorem suppco 8148

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 8150. (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 7871	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
2		simpl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝑍 ∈ V)
3		suppimacnv 8116	. . . . 5 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
4	1, 2, 3	syl2an2 686	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
5		cnvco 5834	. . . . . 6 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
6	5	imaeq1i 6016	. . . . 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 6209	. . . . 5 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
9		simprl 770	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝐹 ∈ 𝑉)
10		suppimacnv 8116	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
11	9, 2, 10	syl2anc 584	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	11	imaeq2d 6019	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
13	8, 12	eqtr4id 2790	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
14	4, 7, 13	3eqtrd 2775	. . 3 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
15	14	ex 412	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))))
16		prcnel 3466	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
17	16	intnand 488	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
18		supp0prc 8105	. . . . 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 488	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21		supp0prc 8105	. . . . . . 7 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
22	20, 21	syl 17	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
23	22	imaeq2d 6019	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
24		ima0 6036	. . . . 5 ⊢ (◡𝐺 “ ∅) = ∅
25	23, 24	eqtrdi 2787	. . . 4 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
26	19, 25	eqtr4d 2774	. . 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 182	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∖ cdif 3898  ∅c0 4285  {csn 4580  ^◡ccnv 5623   “ cima 5627   ∘ ccom 5628  (class class class)co 7358   supp csupp 8102

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8103

This theorem is referenced by:  supp0cosupp0  8150  imacosupp  8151  extvfvcl  33701