Theorem ressupprn 31024

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 6464	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 215	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	3ad2ant1 1132	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		dmexg 7750	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
5	4	3ad2ant2 1133	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → dom 𝐹 ∈ V)
6		simp3 1137	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 0 ∈ 𝑊)
7		elsuppfn 7987	. . . . . . 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 1370	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 )))
9	8	anbi1d 630	. . . . 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 469	. . . . . 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 247	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 )))
13	12	impl 456	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 ))
14	13	fvresd 6794	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
15	14	eqeq1d 2740	. . . . . . . 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 461	. . . . . . . 8 ⊢ (((𝐹‘𝑥) ≠ 0 ∧ (𝐹‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) = 𝑦 ∧ (𝐹‘𝑥) ≠ 0 ))
18		simpr 485	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) = 𝑦)
19	18	neeq1d 3003	. . . . . . . . 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	syl5bb 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 278	. . . . . 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 3224	. . 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 7993	. . . . 5 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
27		fnssres 6555	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
28	3, 26, 27	sylancl 586	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29		fvelrnb 6830	. . . 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 6830	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
32	31	anbi1d 630	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
33		eldifsn 4720	. . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ))
34		r19.41v 3276	. . . . 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 2736	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  {csn 4561  dom cdm 5589  ran crn 5590   ↾ cres 5591  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  (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-rep 5209  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-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  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-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  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-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978

This theorem is referenced by:  fsupprnfi  31026