Theorem suppofss1dcl 6498

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 5532	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		suppofssd.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffnd 5532	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
5		suppofssd.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 3440	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		eqidd 2239	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
8		eqidd 2239	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
9		oveq2 6087	. . . . . . . . 9 ⊢ (𝑣 = (𝐺‘𝑦) → ((𝐹‘𝑦)𝑋𝑣) = ((𝐹‘𝑦)𝑋(𝐺‘𝑦)))
10	9	eleq1d 2307	. . . . . . . 8 ⊢ (𝑣 = (𝐺‘𝑦) → (((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵))
11		oveq1 6086	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑦) → (𝑢𝑋𝑣) = ((𝐹‘𝑦)𝑋𝑣))
12	11	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑦) → ((𝑢𝑋𝑣) ∈ 𝐵 ↔ ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵))
13	12	ralbidv 2550	. . . . . . . . 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 2632	. . . . . . . . . 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 5837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
18	13, 16, 17	rspcdva 2934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∀𝑣 ∈ 𝐵 ((𝐹‘𝑦)𝑋𝑣) ∈ 𝐵)
19	3	ffvelcdmda 5837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐵)
20	10, 18, 19	rspcdva 2934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦)𝑋(𝐺‘𝑦)) ∈ 𝐵)
21	2, 4, 5, 5, 6, 7, 8, 20	ofvalg 6306	. . . . . 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 6094	. . . . 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 2623	. . . . . . . 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 6095	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺‘𝑦)))
30	29	eqeq1d 2247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = (𝐺‘𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺‘𝑦)) = 𝑍))
31	19, 30	rspcdv 2932	. . . . . . 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 2275	. . . 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 2623	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = 𝑍 → ((𝐹 ∘𝑓 𝑋𝐺)‘𝑦) = 𝑍))
37	14, 1, 3, 5, 5, 6	off 6309	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺):𝐴⟶𝐵)
38	37	ffnd 5532	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑋𝐺) Fn 𝐴)
39		ssidd 3269	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
40		suppofssd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
41		suppfnss 6491	. . 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 1285	. 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 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220   Fn wfn 5370  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079   ∘_𝑓 cof 6294   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-supp 6470

This theorem is referenced by: (None)