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Theorem supppreima 30454
 Description: Express the support of a function as the preimage of its range except zero. (Contributed by Thierry Arnoux, 24-Jun-2024.)
Assertion
Ref Expression
supppreima ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (ran 𝐹 ∖ {𝑍})))

Proof of Theorem supppreima
StepHypRef Expression
1 cnvimarndm 5921 . . . 4 (𝐹 “ ran 𝐹) = dom 𝐹
21a1i 11 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 “ ran 𝐹) = dom 𝐹)
32difeq1d 4052 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝐹 “ ran 𝐹) ∖ (𝐹 “ {𝑍})) = (dom 𝐹 ∖ (𝐹 “ {𝑍})))
4 difpreima 6816 . . 3 (Fun 𝐹 → (𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((𝐹 “ ran 𝐹) ∖ (𝐹 “ {𝑍})))
543ad2ant1 1130 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((𝐹 “ ran 𝐹) ∖ (𝐹 “ {𝑍})))
6 suppssdm 7830 . . . 4 (𝐹 supp 𝑍) ⊆ dom 𝐹
7 dfss4 4188 . . . 4 ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
86, 7mpbi 233 . . 3 (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)
9 suppiniseg 30449 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
109difeq2d 4053 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (dom 𝐹 ∖ (𝐹 “ {𝑍})))
118, 10syl5eqr 2850 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (dom 𝐹 ∖ (𝐹 “ {𝑍})))
123, 5, 113eqtr4rd 2847 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (ran 𝐹 ∖ {𝑍})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ∖ cdif 3881   ⊆ wss 3884  {csn 4528  ◡ccnv 5522  dom cdm 5523  ran crn 5524   “ cima 5526  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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