Theorem supppreima 31707

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 6054	. . . 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 4101	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍})))
4		difpreima 7035	. . 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 8128	. . . 4 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
7		dfss4 4238	. . . 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 31702	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
10	9	difeq2d 4102	. . 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 1087   = wceq 1541   ∈ wcel 2106   ∖ cdif 3925   ⊆ wss 3928  {csn 4606  ^◡ccnv 5652  dom cdm 5653  ran crn 5654   “ cima 5656  Fun wfun 6510  (class class class)co 7377   supp csupp 8112

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-supp 8113

This theorem is referenced by:  mptiffisupp  31709  gsumhashmul  32002