Theorem ressupprn 32894

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 6553	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 218	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	3ad2ant1 1147	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		dmexg 7884	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
5	4	3ad2ant2 1148	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → dom 𝐹 ∈ V)
6		simp3 1152	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 0 ∈ 𝑊)
7		elsuppfn 8152	. . . . . . 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 1392	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 )))
9	8	anbi1d 640	. . . . 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 472	. . . . . 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 250	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 )))
13	12	impl 459	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 ))
14	13	fvresd 6889	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
15	14	eqeq1d 2766	. . . . . . . 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 587	. . . . . . 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 464	. . . . . . . 8 ⊢ (((𝐹‘𝑥) ≠ 0 ∧ (𝐹‘𝑥) = 𝑦) ↔ ((𝐹‘𝑥) = 𝑦 ∧ (𝐹‘𝑥) ≠ 0 ))
18		simpr 488	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) = 𝑦)
19	18	neeq1d 3018	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹‘𝑥) ≠ 0 ↔ 𝑦 ≠ 0 ))
20	19	pm5.32da 587	. . . . . . . 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 285	. . . . . . 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 281	. . . . . 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 587	. . . . 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 307	. . . 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 3184	. . 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 8159	. . . . 5 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
27		fnssres 6646	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
28	3, 26, 27	sylancl 595	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29		fvelrnb 6929	. . . 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 6929	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
32	31	anbi1d 640	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 )))
33		eldifsn 4748	. . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ))
34		r19.41v 3194	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ))
35	32, 33, 34	3bitr4g 316	. . . 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 313	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
38	37	eqrdv 2762	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  Vcvv 3456   ∖ cdif 3903   ⊆ wss 3906  {csn 4584  dom cdm 5649  ran crn 5650   ↾ cres 5651  Fun wfun 6517   Fn wfn 6518  ‘cfv 6523  (class class class)co 7398   supp csupp 8142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-supp 8143

This theorem is referenced by:  fsupprnfi  32896