Theorem supppreima 33266

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 6077	. . . 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 4073	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
4		difpreima 7056	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
5	4	3ad2ant1 1151	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
6		suppssdm 8178	. . . 4 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
7		dfss4 4215	. . . 4 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
8	6, 7	mpbi 233	. . 3 ⊢ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)
9		suppiniseg 33261	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
10	9	difeq2d 4074	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
11	8, 10	eqtr3id 2810	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
12	3, 5, 11	3eqtr4rd 2807	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍})))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654  Fun wfun 6525  (class class class)co 7412   supp csupp 8161

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8162

This theorem is used by:  mptiffisupp  33268  gsumhashmul  33610  psrbasfsupp  34125