Theorem suppofss1dcl 6463

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)
suppofss1dcl.cl	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢𝑋𝑣) ∈ 𝐵)
suppofss1d.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)

Assertion

Ref	Expression
suppofss1dcl	⊢ (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   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 suppofss1dcl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 5508	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 5508	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 3429	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9		oveq2 6057	. . . . . . . . 9 ⊢ (𝑣 = (𝐺‘𝑦) → ((𝐹‘𝑦)𝑋𝑣) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	eleq1d 2301	. . . . . . . 8 ⊢ (𝑣 = (𝐺‘𝑦) → (((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵))
11		oveq1 6056	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑦) → (𝑢𝑋𝑣) = ((𝐹‘𝑦)𝑋𝑣))
12	11	eleq1d 2301	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑦) → ((𝑢𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵))
13	12	ralbidv 2542	. . . . . . . . 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 2624	. . . . . . . . . 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 5811	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
18	13, 16, 17	rspcdva 2925	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵)
19	3	ffvelcdmda 5811	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐵)
20	10, 18, 19	rspcdva 2925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵)
21	2, 4, 5, 5, 6, 7, 8, 20	ofvalg 6275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
22	21	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
23		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → (𝐹‘𝑦) = 𝑍)
24	23	oveq1d 6064	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) = (𝑍𝑋(𝐺‘𝑦)))
25		suppofss1d.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)
26	25	ralrimiva 2615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍)
27	26	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍)
28		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → 𝑥 = (𝐺‘𝑦))
29	28	oveq2d 6065	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺‘𝑦)))
30	29	eqeq1d 2241	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺‘𝑦)) = 𝑍))
31	19, 30	rspcdv 2923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺‘𝑦)) = 𝑍))
32	27, 31	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑍𝑋(𝐺‘𝑦)) = 𝑍)
33	32	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → (𝑍𝑋(𝐺‘𝑦)) = 𝑍)
34	22, 24, 33	3eqtrd 2269	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍)
35	34	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍))
36	35	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍))
37	14, 1, 3, 5, 5, 6	off 6278	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺):𝐴⟶𝐵)
38	37	ffnd 5508	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺) Fn 𝐴)
39		ssidd 3258	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
40		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
41		suppfnss 6456	. . 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)))
42	38, 2, 39, 5, 40, 41	syl23anc 1281	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
43	36, 42	mpd 13	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3210   Fn wfn 5346  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049   ∘_𝑓 cof 6263   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-supp 6435

This theorem is referenced by: (None)