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Theorem fressupp 30451
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 6345 . . . . 5 (Fun 𝐹 → Rel 𝐹)
213ad2ant1 1130 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → Rel 𝐹)
3 suppssdm 7830 . . . . 5 (𝐹 supp 𝑍) ⊆ dom 𝐹
4 undif 4391 . . . . . . 7 ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
54biimpi 219 . . . . . 6 ((𝐹 supp 𝑍) ⊆ dom 𝐹 → ((𝐹 supp 𝑍) ∪ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = dom 𝐹)
65eqcomd 2807 . . . . 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 4382 . . . . 5 ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅
98a1i 11 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = ∅)
10 reldisjun 30369 . . . 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 1368 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 = ((𝐹 ↾ (𝐹 supp 𝑍)) ∪ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))))
1211difeq1d 4052 . 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 5847 . . . . 5 (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹
14 sseqin2 4145 . . . . 5 ((𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ 𝐹 ↔ (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
1513, 14mpbi 233 . . . 4 (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))
16 suppiniseg 30449 . . . . . 6 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
1716reseq2d 5822 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ↾ (𝐹 “ {𝑍})))
18 cnvrescnv 6023 . . . . . . 7 (𝐹 ↾ {𝑍}) = (𝐹 ∩ (V × {𝑍}))
19 funcnvres2 6408 . . . . . . 7 (Fun 𝐹(𝐹 ↾ {𝑍}) = (𝐹 ↾ (𝐹 “ {𝑍})))
2018, 19syl5eqr 2850 . . . . . 6 (Fun 𝐹 → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
21203ad2ant1 1130 . . . . 5 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (V × {𝑍})) = (𝐹 ↾ (𝐹 “ {𝑍})))
2217, 21eqtr4d 2839 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 ∩ (V × {𝑍})))
2315, 22syl5eq 2848 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})))
24 indifbi 30295 . . 3 ((𝐹 ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∩ (V × {𝑍})) ↔ (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
2523, 24sylib 221 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ∖ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ∖ (V × {𝑍})))
268reseq2i 5819 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = (𝐹 ↾ ∅)
27 resindi 5838 . . . 4 (𝐹 ↾ ((𝐹 supp 𝑍) ∩ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍))))
28 res0 5826 . . . 4 (𝐹 ↾ ∅) = ∅
2926, 27, 283eqtr3i 2832 . . 3 ((𝐹 ↾ (𝐹 supp 𝑍)) ∩ (𝐹 ↾ (dom 𝐹 ∖ (𝐹 supp 𝑍)))) = ∅
30 undif5 30294 . . 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 2845 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528   × cxp 5521  ccnv 5522  dom cdm 5523  cres 5525  cima 5526  Rel wrel 5528  Fun wfun 6322  (class class class)co 7139   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818
This theorem is referenced by:  gsumhashmul  30744
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