Theorem suppofss2d 8145

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 6657	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 6657	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4180	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9	2, 4, 5, 5, 6, 7, 8	ofval 7628	. . . . . 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 7369	. . . . 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 3121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑥𝑋𝑍) = 𝑍)
16	1	ffvelcdmda 7022	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
17		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → 𝑥 = (𝐹‘𝑦))
18	17	oveq1d 7368	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝑥𝑋𝑍) = ((𝐹‘𝑦)𝑋𝑍))
19	18	eqeq1d 2731	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹‘𝑦)𝑋𝑍) = 𝑍))
20	16, 19	rspcdv 3571	. . . . . . 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 2768	. . . 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 3121	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐺‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
26	2, 4, 5, 5, 6	offn 7630	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺) Fn 𝐴)
27		ssidd 3961	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
28		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29		suppfnss 8129	. . 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 1379	. 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 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   ∘_f cof 7615   supp csupp 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-supp 8101

This theorem is referenced by:  frlmphl  21706