Theorem suppcoss 8201

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 8196, 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 6718	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
4		suppcoss.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
5	3, 4	fcod 6731	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ran 𝐹)
6		eldif 3914	. . . . 5 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
7	4	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
8		suppcoss.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		suppcoss.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		elsuppfn 8164	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑌 ∈ 𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
11	7, 8, 9, 10	syl3anc 1397	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
12	11	notbid 321	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
13	12	anbi2d 641	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌))))
14		annotanannot 847	. . . . . 6 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌))
15	13, 14	bitrdi 290	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
16	6, 15	bitrid 286	. . . 4 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
17		nne 2961	. . . . . 6 ⊢ (¬ (𝐺‘𝑘) ≠ 𝑌 ↔ (𝐺‘𝑘) = 𝑌)
18	17	anbi2i 634	. . . . 5 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌))
19	4	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝐺:𝐵⟶𝐴)
20		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝑘 ∈ 𝐵)
21	19, 20	fvco3d 6982	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘)))
22		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐺‘𝑘) = 𝑌)
23	22	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘𝑌))
24		suppcoss.1	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)
25	24	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘𝑌) = 𝑍)
26	21, 23, 25	3eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
27	26	ex 417	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
28	18, 27	biimtrid 245	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
29	16, 28	sylbid 243	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
30	29	imp 411	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
31	5, 30	suppss 8188	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957   ∖ cdif 3901   ⊆ wss 3904  ran crn 5661   ∘ ccom 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   supp csupp 8154

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8155

This theorem is used by:  mplsubglem  22159  mhpinvcl  22326