Theorem suppcoss 8212

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 8207, 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 6725	. . . 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 6738	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ran 𝐹)
6		eldif 3918	. . . . 5 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
7	4	ffnd 6713	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
8		suppcoss.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		suppcoss.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		elsuppfn 8175	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑌 ∈ 𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
11	7, 8, 9, 10	syl3anc 1398	. . . . . . . 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 642	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌))))
14		annotanannot 848	. . . . . 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 2965	. . . . . 6 ⊢ (¬ (𝐺‘𝑘) ≠ 𝑌 ↔ (𝐺‘𝑘) = 𝑌)
18	17	anbi2i 635	. . . . 5 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌))
19	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝐺:𝐵⟶𝐴)
20		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝑘 ∈ 𝐵)
21	19, 20	fvco3d 6989	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘)))
22		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐺‘𝑘) = 𝑌)
23	22	fveq2d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘𝑌))
24		suppcoss.1	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)
25	24	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘𝑌) = 𝑍)
26	21, 23, 25	3eqtrd 2805	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
27	26	ex 418	. . . . 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 412	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
31	5, 30	suppss 8199	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2961   ∖ cdif 3905   ⊆ wss 3908  ran crn 5667   ∘ ccom 5670   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   supp csupp 8165

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-supp 8166

This theorem is used by:  mplsubglem  22185  mhpinvcl  22352