Theorem suppofss2dcl 6443

Description: Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)

Hypotheses

Ref	Expression
suppofssd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppofssd.2	⊢ (𝜑 → 𝑍 ∈ 𝐵)
suppofssd.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppofssd.4	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
suppofss1dcl.cl	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢𝑋𝑣) ∈ 𝐵)
suppofss2d.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)

Assertion

Ref	Expression
suppofss2dcl	⊢ (𝜑 → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝑥,𝑋   𝑥,𝑍   𝜑,𝑥   𝑢,𝐵,𝑣   𝜑,𝑢,𝑣   𝑢,𝐹,𝑣   𝑣,𝐺   𝑢,𝑋,𝑣

Allowed substitution hints:   𝐴(𝑣,𝑢)   𝐺(𝑢)   𝑉(𝑥,𝑣,𝑢)   𝑍(𝑣,𝑢)

Proof of Theorem suppofss2dcl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 5490	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 5490	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 3418	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9		oveq2 6036	. . . . . . . . 9 ⊢ (𝑣 = (𝐺‘𝑦) → ((𝐹‘𝑦)𝑋𝑣) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	eleq1d 2300	. . . . . . . 8 ⊢ (𝑣 = (𝐺‘𝑦) → (((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵))
11		oveq1 6035	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑦) → (𝑢𝑋𝑣) = ((𝐹‘𝑦)𝑋𝑣))
12	11	eleq1d 2300	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑦) → ((𝑢𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵))
13	12	ralbidv 2533	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑦) → (∀𝑣 ∈ 𝐵 (𝑢𝑋𝑣) ∈ 𝐵 ↔ ∀𝑣 ∈ 𝐵 ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵))
14		suppofss1dcl.cl	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢𝑋𝑣) ∈ 𝐵)
15	14	ralrimivva 2615	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢𝑋𝑣) ∈ 𝐵)
16	15	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢𝑋𝑣) ∈ 𝐵)
17	1	ffvelcdmda 5790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
18	13, 16, 17	rspcdva 2916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵)
19	3	ffvelcdmda 5790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐵)
20	10, 18, 19	rspcdva 2916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵)
21	2, 4, 5, 5, 6, 7, 8, 20	ofvalg 6254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
22	21	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
23		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = 𝑍)
24	23	oveq2d 6044	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) = ((𝐹‘𝑦)𝑋𝑍))
25		suppofss2d.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)
26	25	ralrimiva 2606	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
27	26	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
28		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → 𝑥 = (𝐹‘𝑦))
29	28	oveq1d 6043	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝑥𝑋𝑍) = ((𝐹‘𝑦)𝑋𝑍))
30	29	eqeq1d 2240	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
31	17, 30	rspcdv 2914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍 → ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
32	27, 31	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦)𝑋𝑍) = 𝑍)
33	32	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋𝑍) = 𝑍)
34	22, 24, 33	3eqtrd 2268	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍)
35	34	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍))
36	35	ralrimiva 2606	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍))
37	14, 1, 3, 5, 5, 6	off 6257	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺):𝐴⟶𝐵)
38	37	ffnd 5490	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺) Fn 𝐴)
39		ssidd 3249	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
40		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
41		suppfnss 6435	. . 3 ⊢ ((((𝐹 ∘𝑓 𝑋𝐺) Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵)) → (∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)))
42	38, 4, 39, 5, 40, 41	syl23anc 1281	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)))
43	36, 42	mpd 13	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ⊆ wss 3201   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ∘_𝑓 cof 6242   supp csupp 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-supp 6414

This theorem is referenced by: (None)