Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fressupp Structured version   Visualization version   GIF version

Theorem fressupp 32703
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 6585 . . . . 5 (Fun 𝐹 → Rel 𝐹)
213ad2ant1 1132 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → Rel 𝐹)
3 suppssdm 8201 . . . . 5 (𝐹 supp 𝑍) ⊆ dom 𝐹
4 undif 4488 . . . . . . 7 ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
54biimpi 216 . . . . . 6 ((𝐹 supp 𝑍) ⊆ dom 𝐹 → ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
65eqcomd 2741 . . . . 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 4478 . . . . 5 ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
98a1i 11 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅)
10 reldisjun 6052 . . . 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 1370 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
1211difeq1d 4135 . 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 6022 . . . . 5 (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
14 sseqin2 4231 . . . . 5 ((𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹 ↔ (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
1513, 14mpbi 230 . . . 4 (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
16 suppiniseg 32701 . . . . . 6 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
1716reseq2d 6000 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (𝐹 “ {𝑍})))
18 cnvrescnv 6217 . . . . . . 7 (𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
19 funcnvres2 6648 . . . . . . 7 (Fun 𝐹(𝐹 ↾ {𝑍}) = (𝐹 ↾ (𝐹 “ {𝑍})))
2018, 19eqtr3id 2789 . . . . . 6 (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
21203ad2ant1 1132 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
2217, 21eqtr4d 2778 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
2315, 22eqtrid 2787 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
24 indifbi 32548 . . 3 ((𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})) ↔ (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
2523, 24sylib 218 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
268reseq2i 5997 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
27 resindi 6016 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
28 res0 6004 . . . 4 (𝐹 ↾ ∅) = ∅
2926, 27, 283eqtr3i 2771 . . 3 ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
30 undif5 4491 . . 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 2784 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631   × cxp 5687  ccnv 5688  dom cdm 5689  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  gsumhashmul  33047
  Copyright terms: Public domain W3C validator