Theorem fsuppcor 8869

Description: The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppcor.0	⊢ (𝜑 → 0 ∈ 𝑊)
fsuppcor.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
fsuppcor.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
fsuppcor.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
fsuppcor.s	⊢ (𝜑 → 𝐶 ⊆ 𝐵)
fsuppcor.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
fsuppcor.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fsuppcor.n	⊢ (𝜑 → 𝐹 finSupp 𝑍)
fsuppcor.i	⊢ (𝜑 → (𝐺‘𝑍) = 0 )

Assertion

Ref	Expression
fsuppcor	⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Proof of Theorem fsuppcor

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppcor.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
2	1	ffund 6520	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppcor.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffund 6520	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6397	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 586	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppcor.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 8842	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fsuppcor.s	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
10	1, 9	fssresd 6547	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷)
11		fco2 6535	. . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
12	10, 3, 11	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷)
13		eldifi 4105	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 6762	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	3, 13, 14	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssidd 3992	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17		fsuppcor.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
18		fsuppcor.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	3, 16, 17, 18	suppssr 7863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
20	19	fveq2d 6676	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
21		fsuppcor.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 )
22	21	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 )
23	15, 20, 22	3eqtrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 )
24	12, 23	suppss 7862	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
25	8, 24	ssfid 8743	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)
26		fsuppcor.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
27		fex 6991	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐷 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
28	1, 26, 27	syl2anc 586	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
29		fex 6991	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V)
30	3, 17, 29	syl2anc 586	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
31		coexg 7636	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
32	28, 30, 31	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
33		fsuppcor.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝑊)
34		isfsupp 8839	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
35	32, 33, 34	syl2anc 586	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin)))
36	6, 25, 35	mpbir2and 711	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938   class class class wbr 5068   ↾ cres 5559   ∘ ccom 5561  Fun wfun 6351  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   supp csupp 7832  Fincfn 8511   finSupp cfsupp 8835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-supp 7833  df-er 8291  df-en 8512  df-fin 8515  df-fsupp 8836

This theorem is referenced by:  mapfienlem1  8870  mapfienlem2  8871  cpmadumatpolylem2  21492  selvval2lem4  39143