Theorem suppofss2d 7852

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 6488	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 6488	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4145	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9	2, 4, 5, 5, 6, 7, 8	ofval 7398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
11		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → (𝐺‘𝑦) = 𝑍)
12	11	oveq2d 7151	. . . . 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 3149	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
15	14	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
16	1	ffvelrnda 6828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
17		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → 𝑥 = (𝐹‘𝑦))
18	17	oveq1d 7150	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝑥𝑋𝑍) = ((𝐹‘𝑦)𝑋𝑍))
19	18	eqeq1d 2800	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
20	16, 19	rspcdv 3563	. . . . . . 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 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋𝑍) = 𝑍)
23	10, 12, 22	3eqtrd 2837	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐺‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍)
24	23	ex 416	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
25	24	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
26	2, 4, 5, 5, 6	offn 7400	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺) Fn 𝐴)
27		ssidd 3938	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
28		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29		suppfnss 7838	. . 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 1374	. 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387   supp csupp 7813

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-supp 7814

This theorem is referenced by:  frlmphl  20470