Theorem suppco 8158

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 8160. (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 7881	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
2		simpl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝑍 ∈ V)
3		suppimacnv 8126	. . . . 5 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
4	1, 2, 3	syl2an2 687	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
5		cnvco 5842	. . . . . 6 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
6	5	imaeq1i 6024	. . . . 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 6217	. . . . 5 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
9		simprl 771	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → 𝐹 ∈ 𝑉)
10		suppimacnv 8126	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
11	9, 2, 10	syl2anc 585	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	11	imaeq2d 6027	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
13	8, 12	eqtr4id 2791	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
14	4, 7, 13	3eqtrd 2776	. . 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 3468	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
17	16	intnand 488	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
18		supp0prc 8115	. . . . 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 8115	. . . . . . 7 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
22	20, 21	syl 17	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
23	22	imaeq2d 6027	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
24		ima0 6044	. . . . 5 ⊢ (◡𝐺 “ ∅) = ∅
25	23, 24	eqtrdi 2788	. . . 4 ⊢ (¬ 𝑍 ∈ V → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
26	19, 25	eqtr4d 2775	. . 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 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900  ∅c0 4287  {csn 4582  ^◡ccnv 5631   “ cima 5635   ∘ ccom 5636  (class class class)co 7368   supp csupp 8112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-supp 8113

This theorem is referenced by:  supp0cosupp0  8160  imacosupp  8161  extvfvcl  33712