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Theorem fressupp 32177
Description: The restriction of a function to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)
Assertion
Ref Expression
fressupp ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))

Proof of Theorem fressupp
StepHypRef Expression
1 funrel 6564 . . . . 5 (Fun 𝐹 → Rel 𝐹)
213ad2ant1 1131 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → Rel 𝐹)
3 suppssdm 8164 . . . . 5 (𝐹 supp 𝑍) ⊆ dom 𝐹
4 undif 4480 . . . . . . 7 ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
54biimpi 215 . . . . . 6 ((𝐹 supp 𝑍) ⊆ dom 𝐹 → ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
65eqcomd 2736 . . . . 5 ((𝐹 supp 𝑍) ⊆ dom 𝐹 → dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
73, 6mp1i 13 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
8 disjdif 4470 . . . . 5 ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
98a1i 11 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅)
10 reldisjun 6031 . . . 4 ((Rel 𝐹 ∧ dom 𝐹 = ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ∧ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
112, 7, 9, 10syl3anc 1369 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
1211difeq1d 4120 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
13 resss 6005 . . . . 5 (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
14 sseqin2 4214 . . . . 5 ((𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹 ↔ (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
1513, 14mpbi 229 . . . 4 (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
16 suppiniseg 32175 . . . . . 6 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
1716reseq2d 5980 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (𝐹 “ {𝑍})))
18 cnvrescnv 6193 . . . . . . 7 (𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
19 funcnvres2 6627 . . . . . . 7 (Fun 𝐹(𝐹 ↾ {𝑍}) = (𝐹 ↾ (𝐹 “ {𝑍})))
2018, 19eqtr3id 2784 . . . . . 6 (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
21203ad2ant1 1131 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
2217, 21eqtr4d 2773 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
2315, 22eqtrid 2782 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
24 indifbi 32025 . . 3 ((𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})) ↔ (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
2523, 24sylib 217 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
268reseq2i 5977 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
27 resindi 5996 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
28 res0 5984 . . . 4 (𝐹 ↾ ∅) = ∅
2926, 27, 283eqtr3i 2766 . . 3 ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
30 undif5 4483 . . 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)))
3129, 30mp1i 13 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (𝐹 supp 𝑍)))
3212, 25, 313eqtr3rd 2779 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1539  wcel 2104  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627   × cxp 5673  ccnv 5674  dom cdm 5675  cres 5677  cima 5678  Rel wrel 5680  Fun wfun 6536  (class class class)co 7411   supp csupp 8148
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8149
This theorem is referenced by:  gsumhashmul  32478
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