Theorem fsuppcorn 7306

Description: The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the (𝐹 supp 𝑍) ⊆ ran 𝐺 condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7250. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)

Hypotheses

Ref	Expression
fsuppco.f	⊢ (𝜑 → 𝐹 finSupp 𝑍)
fsuppco.g	⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)
fsuppco.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fsuppco.v	⊢ (𝜑 → 𝐹 ∈ 𝑉)
fsuppcorn.g	⊢ (𝜑 → 𝐺 ∈ 𝑈)
fsuppcorn.rn	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ran 𝐺)

Assertion

Ref	Expression
fsuppcorn	⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)

Proof of Theorem fsuppcorn

Step	Hyp	Ref	Expression
1		fsuppco.v	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		fsuppco.g	. . . 4 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌)
3		df-f1 5385	. . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺))
4	3	simprbi 275	. . . 4 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺)
5	2, 4	syl 14	. . 3 ⊢ (𝜑 → Fun ◡𝐺)
6		cofunex2g 6343	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V)
7	1, 5, 6	syl2anc 415	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V)
8		fsuppco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
9		fsuppco.f	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
10	9	fsuppfund 7299	. . 3 ⊢ (𝜑 → Fun 𝐹)
11		f1fun 5604	. . . 4 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺)
12	2, 11	syl 14	. . 3 ⊢ (𝜑 → Fun 𝐺)
13		funco 5420	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
14	10, 12, 13	syl2anc 415	. 2 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺))
15		fsuppcorn.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑈)
16		suppcofn 6510	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑈) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
17	1, 15, 10, 12, 16	syl22anc 1279	. . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
18	9	fsuppimpd 7298	. . . 4 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
19		f1cnv 5666	. . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 → ◡𝐺:ran 𝐺–1-1-onto→𝑋)
20	2, 19	syl 14	. . . . . 6 ⊢ (𝜑 → ◡𝐺:ran 𝐺–1-1-onto→𝑋)
21		f1of1 5641	. . . . . 6 ⊢ (◡𝐺:ran 𝐺–1-1-onto→𝑋 → ◡𝐺:ran 𝐺–1-1→𝑋)
22	20, 21	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐺:ran 𝐺–1-1→𝑋)
23		fsuppcorn.rn	. . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ran 𝐺)
24		f1imaeng 7083	. . . . 5 ⊢ ((◡𝐺:ran 𝐺–1-1→𝑋 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺 ∧ (𝐹 supp 𝑍) ∈ Fin) → (◡𝐺 “ (𝐹 supp 𝑍)) ≈ (𝐹 supp 𝑍))
25	22, 23, 18, 24	syl3anc 1278	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) ≈ (𝐹 supp 𝑍))
26		enfii 7180	. . . 4 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (◡𝐺 “ (𝐹 supp 𝑍)) ≈ (𝐹 supp 𝑍)) → (◡𝐺 “ (𝐹 supp 𝑍)) ∈ Fin)
27	18, 25, 26	syl2anc 415	. . 3 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) ∈ Fin)
28	17, 27	eqeltrd 2315	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)
29	7, 8, 14, 28	isfsuppd 7295	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   class class class wbr 4131  ^◡ccnv 4775  ran crn 4777   “ cima 4779   ∘ ccom 4780  Fun wfun 5374  ⟶wf 5376  –1-1→wf1 5377  –1-1-onto→wf1o 5379  (class class class)co 6088   supp csupp 6479   ≈ cen 7024  Fincfn 7026   finSupp cfsupp 7290

This proof depends on 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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480  df-er 6811  df-en 7027  df-fin 7029  df-fsupp 7291

This theorem is used by: (None)