Theorem ressupprn 31605

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 6531	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 215	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	3ad2ant1 1133	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
4		dmexg 7840	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
5	4	3ad2ant2 1134	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → dom 𝐹 ∈ V)
6		simp3 1138	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → 0 ∈ 𝑊)
7		elsuppfn 8102	. . . . . . 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 1371	. . . . . 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 6862	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹‘𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
15	14	eqeq1d 2738	. . . . . . . 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	bitrid 282	. . . . . . 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 304	. . . 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 3171	. . 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 8108	. . . . 5 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
27		fnssres 6624	. . . . 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 6903	. . . 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 6903	. . . . . 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 4747	. . . . 5 ⊢ (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑦 ≠ 0 ))
34		r19.41v 3185	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹((𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦 ∧ 𝑦 ≠ 0 ))
35	32, 33, 34	3bitr4g 313	. . . 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 310	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
38	37	eqrdv 2734	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  {csn 4586  dom cdm 5633  ran crn 5634   ↾ cres 5635  Fun wfun 6490   Fn wfn 6491  ‘cfv 6496  (class class class)co 7357   supp csupp 8092

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-supp 8093

This theorem is referenced by:  fsupprnfi  31607