Theorem suppcoss 8232

Description: The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8227, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.)

Hypotheses

Ref	Expression
suppcoss.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
suppcoss.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
suppcoss.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
suppcoss.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
suppcoss.1	⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)

Assertion

Ref	Expression
suppcoss	⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Proof of Theorem suppcoss

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppcoss.f	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		dffn3 6748	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
4		suppcoss.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
5	3, 4	fcod 6761	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ran 𝐹)
6		eldif 3961	. . . . 5 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
7	4	ffnd 6737	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
8		suppcoss.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		suppcoss.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		elsuppfn 8195	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑌 ∈ 𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
11	7, 8, 9, 10	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
12	11	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌))))
14		annotanannot 835	. . . . . 6 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌))
15	13, 14	bitrdi 287	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
16	6, 15	bitrid 283	. . . 4 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
17		nne 2944	. . . . . 6 ⊢ (¬ (𝐺‘𝑘) ≠ 𝑌 ↔ (𝐺‘𝑘) = 𝑌)
18	17	anbi2i 623	. . . . 5 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌))
19	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝐺:𝐵⟶𝐴)
20		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝑘 ∈ 𝐵)
21	19, 20	fvco3d 7009	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘)))
22		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐺‘𝑘) = 𝑌)
23	22	fveq2d 6910	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘𝑌))
24		suppcoss.1	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘𝑌) = 𝑍)
26	21, 23, 25	3eqtrd 2781	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
27	26	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
28	18, 27	biimtrid 242	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
29	16, 28	sylbid 240	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
30	29	imp 406	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
31	5, 30	suppss 8219	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∖ cdif 3948   ⊆ wss 3951  ran crn 5686   ∘ ccom 5689   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   supp csupp 8185

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186

This theorem is referenced by:  mplsubglem  22019  mhpinvcl  22156