Theorem supp0cosupp0 7599

Description: The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)

Assertion

Ref	Expression
supp0cosupp0	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

Proof of Theorem supp0cosupp0

Step	Hyp	Ref	Expression
1		simpl 476	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ 𝑉)
2	1	anim2i 612	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ 𝐹 ∈ 𝑉))
3	2	ancomd 455	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V))
4		suppimacnv 7570	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
5	3, 4	syl 17	. . . . 5 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
6	5	eqeq1d 2827	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) = ∅ ↔ (◡𝐹 “ (V ∖ {𝑍})) = ∅))
7		coexg 7379	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
8	7	anim2i 612	. . . . . . . 8 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑍 ∈ V ∧ (𝐹 ∘ 𝐺) ∈ V))
9	8	ancomd 455	. . . . . . 7 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
10		suppimacnv 7570	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
12		cnvco 5540	. . . . . . . . 9 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
13	12	imaeq1i 5704	. . . . . . . 8 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
14		imaco 5881	. . . . . . . 8 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
15	13, 14	eqtri 2849	. . . . . . 7 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
16		imaeq2 5703	. . . . . . . 8 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) = ∅ → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐺 “ ∅))
17		ima0 5722	. . . . . . . 8 ⊢ (◡𝐺 “ ∅) = ∅
18	16, 17	syl6eq 2877	. . . . . . 7 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) = ∅ → (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))) = ∅)
19	15, 18	syl5eq 2873	. . . . . 6 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) = ∅ → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ∅)
20	11, 19	sylan9eq 2881	. . . . 5 ⊢ (((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) ∧ (◡𝐹 “ (V ∖ {𝑍})) = ∅) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
21	20	ex 403	. . . 4 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((◡𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))
22	6, 21	sylbid 232	. . 3 ⊢ ((𝑍 ∈ V ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))
23	22	ex 403	. 2 ⊢ (𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)))
24		id 22	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V)
25	24	intnand 484	. . . 4 ⊢ (¬ 𝑍 ∈ V → ¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V))
26		supp0prc 7562	. . . 4 ⊢ (¬ ((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
27	25, 26	syl 17	. . 3 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
28	27	2a1d 26	. 2 ⊢ (¬ 𝑍 ∈ V → ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)))
29	23, 28	pm2.61i 177	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ∖ cdif 3795  ∅c0 4144  {csn 4397  ^◡ccnv 5341   “ cima 5345   ∘ ccom 5346  (class class class)co 6905   supp csupp 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-supp 7560

This theorem is referenced by:  gsumval3lem2  18660