Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ressupprn Structured version   Visualization version   GIF version

Theorem ressupprn 32699
Description: The range of a function restricted to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)
Assertion
Ref Expression
ressupprn ((Fun 𝐹𝐹𝑉0𝑊) → 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.
StepHypRef Expression
1 funfn 6596 . . . . . . . . 9 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 216 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
323ad2ant1 1134 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 𝐹 Fn dom 𝐹)
4 dmexg 7923 . . . . . . . 8 (𝐹𝑉 → dom 𝐹 ∈ V)
543ad2ant2 1135 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → dom 𝐹 ∈ V)
6 simp3 1139 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 0𝑊)
7 elsuppfn 8195 . . . . . . 7 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ V ∧ 0𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 )))
83, 5, 6, 7syl3anc 1373 . . . . . 6 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 )))
98anbi1d 631 . . . . 5 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ (𝐹 supp 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)))
10 anass 468 . . . . . 6 (((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)))
1110a1i 11 . . . . 5 ((Fun 𝐹𝐹𝑉0𝑊) → (((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦))))
128biimprd 248 . . . . . . . . . . 11 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 )))
1312impl 455 . . . . . . . . . 10 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 ))
1413fvresd 6926 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
1514eqeq1d 2739 . . . . . . . 8 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝑦))
1615pm5.32da 579 . . . . . . 7 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦)))
17 ancom 460 . . . . . . . 8 (((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦 ∧ (𝐹𝑥) ≠ 0 ))
18 simpr 484 . . . . . . . . . 10 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) = 𝑦)
1918neeq1d 3000 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ≠ 0𝑦0 ))
2019pm5.32da 579 . . . . . . . 8 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) = 𝑦 ∧ (𝐹𝑥) ≠ 0 ) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2117, 20bitrid 283 . . . . . . 7 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2216, 21bitrd 279 . . . . . 6 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2322pm5.32da 579 . . . . 5 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) = 𝑦𝑦0 ))))
249, 11, 233bitrd 305 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ (𝐹 supp 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) = 𝑦𝑦0 ))))
2524rexbidv2 3175 . . 3 ((Fun 𝐹𝐹𝑉0𝑊) → (∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 )))
26 suppssdm 8202 . . . . 5 (𝐹 supp 0 ) ⊆ dom 𝐹
27 fnssres 6691 . . . . 5 ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
283, 26, 27sylancl 586 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29 fvelrnb 6969 . . . 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))
3028, 29syl 17 . . 3 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ ∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦))
31 fvelrnb 6969 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
3231anbi1d 631 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ ran 𝐹𝑦0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦𝑦0 )))
33 eldifsn 4786 . . . . 5 (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹𝑦0 ))
34 r19.41v 3189 . . . . 5 (∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦𝑦0 ))
3532, 33, 343bitr4g 314 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ ∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 )))
363, 35syl 17 . . 3 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ ∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 )))
3725, 30, 363bitr4d 311 . 2 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
3837eqrdv 2735 1 ((Fun 𝐹𝐹𝑉0𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  {csn 4626  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  fsupprnfi  32701
  Copyright terms: Public domain W3C validator