Theorem suppofss2d 8211

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)
suppofss2d.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)

Assertion

Ref	Expression
suppofss2d	⊢ (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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem suppofss2d

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6716	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 6716	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4207	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9	2, 4, 5, 5, 6, 7, 8	ofval 7689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
11		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = 𝑍)
12	11	oveq2d 7428	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) = ((𝐹‘𝑦)𝑋𝑍))
13		suppofss2d.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)
14	13	ralrimiva 3133	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
16	1	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
17		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → 𝑥 = (𝐹‘𝑦))
18	17	oveq1d 7427	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝑥𝑋𝑍) = ((𝐹‘𝑦)𝑋𝑍))
19	18	eqeq1d 2736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
20	16, 19	rspcdv 3597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍 → ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
21	15, 20	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦)𝑋𝑍) = 𝑍)
22	21	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋𝑍) = 𝑍)
23	10, 12, 22	3eqtrd 2773	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍)
24	23	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
25	24	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
26	2, 4, 5, 5, 6	offn 7691	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺) Fn 𝐴)
27		ssidd 3987	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
28		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29		suppfnss 8195	. . 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)))
30	26, 4, 27, 5, 28, 29	syl23anc 1378	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)))
31	25, 30	mpd 15	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3931   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7412   ∘_f cof 7676   supp csupp 8166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7678  df-supp 8167

This theorem is referenced by:  frlmphl  21754