Theorem fressupp 32892

Description: The restriction of a function to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)

Assertion

Ref	Expression
fressupp	⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))

Proof of Theorem fressupp

Step	Hyp	Ref	Expression
1		funrel 6540	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2	1	3ad2ant1 1147	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → Rel 𝐹)
3		suppssdm 8159	. . . . 5 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
4		undif 4438	. . . . . . 7 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
5	4	biimpi 218	. . . . . 6 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 → ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
6	5	eqcomd 2770	. . . . 5 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 → dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
7	3, 6	mp1i 13	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
8		reldmun 6022	. . . 4 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
9	2, 7, 8	syl2anc 593	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
10	9	difeq1d 4081	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
11		resss 5989	. . . . 5 ⊢ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
12		sseqin2 4177	. . . . 5 ⊢ ((𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹 ↔ (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
13	11, 12	mpbi 232	. . . 4 ⊢ (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
14		suppiniseg 32890	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
15	14	reseq2d 5967	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (◡𝐹 “ {𝑍})))
16		cnvrescnv 6184	. . . . . . 7 ⊢ ◡(◡𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
17		funcnvres2 6603	. . . . . . 7 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ {𝑍}) = (𝐹 ↾ (◡𝐹 “ {𝑍})))
18	16, 17	eqtr3id 2813	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (◡𝐹 “ {𝑍})))
19	18	3ad2ant1 1147	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (◡𝐹 “ {𝑍})))
20	15, 19	eqtr4d 2802	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
21	13, 20	eqtrid 2811	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
22		indifbi 32721	. . 3 ⊢ ((𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})) ↔ (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
23	21, 22	sylib 220	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
24		disjdif 4428	. . . . 5 ⊢ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
25	24	reseq2i 5964	. . . 4 ⊢ (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
26		resindi 5983	. . . 4 ⊢ (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
27		res0 5971	. . . 4 ⊢ (𝐹 ↾ ∅) = ∅
28	25, 26, 27	3eqtr3i 2795	. . 3 ⊢ ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
29		undif5 4440	. . 3 ⊢ (((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅ → (((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (𝐹 supp 𝑍)))
30	28, 29	mp1i 13	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (𝐹 supp 𝑍)))
31	10, 23, 30	3eqtr3rd 2808	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4584   × cxp 5647  ^◡ccnv 5648  dom cdm 5649   ↾ cres 5651   “ cima 5652  Rel wrel 5654  Fun wfun 6517  (class class class)co 7398   supp csupp 8142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-supp 8143

This theorem is referenced by:  gsumhashmul  33249