Theorem ressupprn 32751

Description: The range of a function restricted to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)

Assertion

Ref	Expression
ressupprn	⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))

Proof of Theorem ressupprn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6517	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 216	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	3ad2ant1 1134	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		dmexg 7841	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
5	4	3ad2ant2 1135	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → dom 𝐹 ∈ V)
6		simp3 1139	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 0 ∈ 𝑊)
7		elsuppfn 8109	. . . . . . 7 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ V ∧ 0 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 )))
8	3, 5, 6, 7	syl3anc 1374	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 )))
9	8	anbi1d 632	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ (𝐹 supp 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)))
10		anass 468	. . . . . 6 ⊢ (((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)))
11	10	a1i 11	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦))))
12	8	biimprd 248	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 )))
13	12	impl 455	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 ))
14	13	fvresd 6849	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
15	14	eqeq1d 2737	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ (𝐹‘𝑥) = 𝑦))
16	15	pm5.32da 579	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) ≠ 0 ∧ (𝐹‘𝑥) = 𝑦)))
17		ancom 460	. . . . . . . 8 ⊢ (((𝐹‘𝑥) ≠ 0 ∧ (𝐹‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) = 𝑦 ∧ (𝐹‘𝑥) ≠ 0 ))
18		simpr 484	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) = 𝑦)
19	18	neeq1d 2989	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ≠ 0 ↔ 𝑦 ≠ 0 ))
20	19	pm5.32da 579	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) = 𝑦 ∧ (𝐹‘𝑥) ≠ 0 ) ↔ ((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
21	17, 20	bitrid 283	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) ≠ 0 ∧ (𝐹‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
22	16, 21	bitrd 279	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
23	22	pm5.32da 579	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ))))
24	9, 11, 23	3bitrd 305	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ (𝐹 supp 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ))))
25	24	rexbidv2 3155	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
26		suppssdm 8116	. . . . 5 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
27		fnssres 6610	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
28	3, 26, 27	sylancl 587	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29		fvelrnb 6889	. . . 4 ⊢ ((𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦))
30	28, 29	syl 17	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦))
31		fvelrnb 6889	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
32	31	anbi1d 632	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
33		eldifsn 4721	. . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ))
34		r19.41v 3165	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ))
35	32, 33, 34	3bitr4g 314	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ ∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
36	3, 35	syl 17	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ ∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
37	25, 30, 36	3bitr4d 311	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
38	37	eqrdv 2733	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∃wrex 3059  Vcvv 3427   ∖ cdif 3882   ⊆ wss 3885  {csn 4557  dom cdm 5620  ran crn 5621   ↾ cres 5622  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  (class class class)co 7356   supp csupp 8099

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-supp 8100

This theorem is referenced by:  fsupprnfi  32753