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Theorem fressupp 32700
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 6595 . . . . 5 (Fun 𝐹 → Rel 𝐹)
213ad2ant1 1133 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → Rel 𝐹)
3 suppssdm 8218 . . . . 5 (𝐹 supp 𝑍) ⊆ dom 𝐹
4 undif 4505 . . . . . . 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 2746 . . . . 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 4495 . . . . 5 ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
98a1i 11 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅)
10 reldisjun 6061 . . . 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 1371 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
1211difeq1d 4148 . 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 6031 . . . . 5 (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
14 sseqin2 4244 . . . . 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 32698 . . . . . 6 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
1716reseq2d 6009 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (𝐹 “ {𝑍})))
18 cnvrescnv 6226 . . . . . . 7 (𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
19 funcnvres2 6658 . . . . . . 7 (Fun 𝐹(𝐹 ↾ {𝑍}) = (𝐹 ↾ (𝐹 “ {𝑍})))
2018, 19eqtr3id 2794 . . . . . 6 (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
21203ad2ant1 1133 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
2217, 21eqtr4d 2783 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
2315, 22eqtrid 2792 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
24 indifbi 32550 . . 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 6006 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
27 resindi 6025 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
28 res0 6013 . . . 4 (𝐹 ↾ ∅) = ∅
2926, 27, 283eqtr3i 2776 . . 3 ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
30 undif5 4508 . . 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 2789 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1537  wcel 2108  Vcvv 3488  cdif 3973  cun 3974  cin 3975  wss 3976  c0 4352  {csn 4648   × cxp 5698  ccnv 5699  dom cdm 5700  cres 5702  cima 5703  Rel wrel 5705  Fun wfun 6567  (class class class)co 7448   supp csupp 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202
This theorem is referenced by:  gsumhashmul  33040
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