Theorem supppreima 31025

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 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍})))

Proof of Theorem supppreima

Step	Hyp	Ref	Expression
1		cnvimarndm 5990	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2	1	a1i 11	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ ran 𝐹) = dom 𝐹)
3	2	difeq1d 4056	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
4		difpreima 6942	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
5	4	3ad2ant1 1132	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
6		suppssdm 7993	. . . 4 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
7		dfss4 4192	. . . 4 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
8	6, 7	mpbi 229	. . 3 ⊢ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)
9		suppiniseg 31020	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
10	9	difeq2d 4057	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
11	8, 10	eqtr3id 2792	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
12	3, 5, 11	3eqtr4rd 2789	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍})))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ⊆ wss 3887  {csn 4561  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592  Fun wfun 6427  (class class class)co 7275   supp csupp 7977

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978

This theorem is referenced by:  gsumhashmul  31316