Theorem suppofss1d 8245

Description: Condition for the support of a function operation to be a subset of the support of the left 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)
suppofss1d.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)

Assertion

Ref	Expression
suppofss1d	⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 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 suppofss1d

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6748	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 6748	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4248	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9	2, 4, 5, 5, 6, 7, 8	ofval 7725	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
11		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → (𝐹‘𝑦) = 𝑍)
12	11	oveq1d 7463	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) = (𝑍𝑋(𝐺‘𝑦)))
13		suppofss1d.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)
14	13	ralrimiva 3152	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍)
16	3	ffvelcdmda 7118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐵)
17		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → 𝑥 = (𝐺‘𝑦))
18	17	oveq2d 7464	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺‘𝑦)))
19	18	eqeq1d 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺‘𝑦)) = 𝑍))
20	16, 19	rspcdv 3627	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺‘𝑦)) = 𝑍))
21	15, 20	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑍𝑋(𝐺‘𝑦)) = 𝑍)
22	21	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → (𝑍𝑋(𝐺‘𝑦)) = 𝑍)
23	10, 12, 22	3eqtrd 2784	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍)
24	23	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
25	24	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍))
26	2, 4, 5, 5, 6	offn 7727	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺) Fn 𝐴)
27		ssidd 4032	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
28		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29		suppfnss 8230	. . 3 ⊢ ((((𝐹 ∘f 𝑋𝐺) Fn 𝐴 ∧ 𝐹 Fn 𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵)) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
30	26, 2, 27, 5, 28, 29	syl23anc 1377	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
31	25, 30	mpd 15	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-supp 8202

This theorem is referenced by:  frlmphllem  21823  rrxcph  25445