Theorem offinsupp1 32678

Description: Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)

Hypotheses

Ref	Expression
offinsupp1.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offinsupp1.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
offinsupp1.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
offinsupp1.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
offinsupp1.g	⊢ (𝜑 → 𝐺:𝐴⟶𝑇)
offinsupp1.1	⊢ (𝜑 → 𝐹 finSupp 𝑌)
offinsupp1.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍)

Assertion

Ref	Expression
offinsupp1	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍)

Distinct variable groups:   𝑥,𝐺   𝑥,𝑅   𝑥,𝑇   𝑥,𝑌   𝑥,𝑍   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem offinsupp1

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		offinsupp1.1	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑌)
2	1	fsuppimpd 9259	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑌) ∈ Fin)
3		ssidd 3959	. . . 4 ⊢ (𝜑 → (𝐹 supp 𝑌) ⊆ (𝐹 supp 𝑌))
4		offinsupp1.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍)
5		offinsupp1.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
6		offinsupp1.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝑇)
7		offinsupp1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		offinsupp1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
9	3, 4, 5, 6, 7, 8	suppssof1 8132	. . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑌))
10	2, 9	ssfid 9158	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)
11		ovexd 7384	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑆 ∧ 𝑗 ∈ 𝑇)) → (𝑖𝑅𝑗) ∈ V)
12		inidm 4178	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
13	11, 5, 6, 7, 7, 12	off 7631	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐴⟶V)
14	13	ffund 6656	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺))
15		ovexd 7384	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) ∈ V)
16		offinsupp1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
17		funisfsupp 9257	. . 3 ⊢ ((Fun (𝐹 ∘f 𝑅𝐺) ∧ (𝐹 ∘f 𝑅𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin))
18	14, 15, 16, 17	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin))
19	10, 18	mpbird 257	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   class class class wbr 5092  Fun wfun 6476  ⟶wf 6478  (class class class)co 7349   ∘_f cof 7611   supp csupp 8093  Fincfn 8872   finSupp cfsupp 9251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-supp 8094  df-1o 8388  df-en 8873  df-fin 8876  df-fsupp 9252

This theorem is referenced by:  fedgmullem1  33612