Theorem fsuppcorn 7291

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 7236. (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 5377	. . . . 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 6329	. . 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 7284	. . 3 ⊢ (𝜑 → Fun 𝐹)
11		f1fun 5596	. . . 4 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺)
12	2, 11	syl 14	. . 3 ⊢ (𝜑 → Fun 𝐺)
13		funco 5412	. . 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 6496	. . . 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 7283	. . . 4 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
19		f1cnv 5658	. . . . . . 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 5633	. . . . . 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 7069	. . . . 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 7166	. . . 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 7280	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   class class class wbr 4125  ^◡ccnv 4768  ran crn 4770   “ cima 4772   ∘ ccom 4773  Fun wfun 5366  ⟶wf 5368  –1-1→wf1 5369  –1-1-onto→wf1o 5371  (class class class)co 6075   supp csupp 6465   ≈ cen 7010  Fincfn 7012   finSupp cfsupp 7275

This theorem was proved from 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-supp 6466  df-er 6797  df-en 7013  df-fin 7015  df-fsupp 7276

This theorem is referenced by: (None)