Theorem fsuppcorn 7267

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 7212. (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 5362	. . . . 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 6312	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V)
7	1, 5, 6	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V)
8		fsuppco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
9		fsuppco.f	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
10	9	fsuppfund 7260	. . 3 ⊢ (𝜑 → Fun 𝐹)
11		f1fun 5581	. . . 4 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺)
12	2, 11	syl 14	. . 3 ⊢ (𝜑 → Fun 𝐺)
13		funco 5397	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
14	10, 12, 13	syl2anc 411	. 2 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺))
15		fsuppcorn.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑈)
16		suppcofn 6479	. . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑈) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
17	1, 15, 10, 12, 16	syl22anc 1275	. . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
18	9	fsuppimpd 7259	. . . 4 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
19		f1cnv 5643	. . . . . . 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 5618	. . . . . 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 7045	. . . . 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 1274	. . . 4 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) ≈ (𝐹 supp 𝑍))
26		enfii 7142	. . . 4 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ (◡𝐺 “ (𝐹 supp 𝑍)) ≈ (𝐹 supp 𝑍)) → (◡𝐺 “ (𝐹 supp 𝑍)) ∈ Fin)
27	18, 25, 26	syl2anc 411	. . 3 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) ∈ Fin)
28	17, 27	eqeltrd 2311	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)
29	7, 8, 14, 28	isfsuppd 7256	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214   class class class wbr 4114  ^◡ccnv 4753  ran crn 4755   “ cima 4757   ∘ ccom 4758  Fun wfun 5351  ⟶wf 5353  –1-1→wf1 5354  –1-1-onto→wf1o 5356  (class class class)co 6058   supp csupp 6448   ≈ cen 6986  Fincfn 6988   finSupp cfsupp 7251

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449  df-er 6780  df-en 6989  df-fin 6991  df-fsupp 7252

This theorem is referenced by: (None)