Theorem supppreima 32770

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 6042	. . . 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 4077	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
4		difpreima 7010	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
5	4	3ad2ant1 1133	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})))
6		suppssdm 8119	. . . 4 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
7		dfss4 4221	. . . 4 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
8	6, 7	mpbi 230	. . 3 ⊢ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)
9		suppiniseg 32765	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
10	9	difeq2d 4078	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
11	8, 10	eqtr3id 2785	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
12	3, 5, 11	3eqtr4rd 2782	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍})))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898   ⊆ wss 3901  {csn 4580  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486  (class class class)co 7358   supp csupp 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8103

This theorem is referenced by:  mptiffisupp  32772  gsumhashmul  33150  psrbasfsupp  33693