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Theorem fressupp 31910
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 6566 . . . . 5 (Fun 𝐹 → Rel 𝐹)
213ad2ant1 1134 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → Rel 𝐹)
3 suppssdm 8162 . . . . 5 (𝐹 supp 𝑍) ⊆ dom 𝐹
4 undif 4482 . . . . . . 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 2739 . . . . 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 4472 . . . . 5 ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
98a1i 11 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅)
10 reldisjun 6033 . . . 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 1372 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
1211difeq1d 4122 . 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 6007 . . . . 5 (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
14 sseqin2 4216 . . . . 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 31908 . . . . . 6 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
1716reseq2d 5982 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (𝐹 “ {𝑍})))
18 cnvrescnv 6195 . . . . . . 7 (𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
19 funcnvres2 6629 . . . . . . 7 (Fun 𝐹(𝐹 ↾ {𝑍}) = (𝐹 ↾ (𝐹 “ {𝑍})))
2018, 19eqtr3id 2787 . . . . . 6 (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
21203ad2ant1 1134 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
2217, 21eqtr4d 2776 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
2315, 22eqtrid 2785 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
24 indifbi 31758 . . 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 5979 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
27 resindi 5998 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
28 res0 5986 . . . 4 (𝐹 ↾ ∅) = ∅
2926, 27, 283eqtr3i 2769 . . 3 ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
30 undif5 4485 . . 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 2782 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629   × cxp 5675  ccnv 5676  dom cdm 5677  cres 5679  cima 5680  Rel wrel 5682  Fun wfun 6538  (class class class)co 7409   supp csupp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-supp 8147
This theorem is referenced by:  gsumhashmul  32208
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