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Theorem ressupprn 30453
 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 6358 . . . . . . . . 9 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 219 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
323ad2ant1 1130 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 𝐹 Fn dom 𝐹)
4 dmexg 7598 . . . . . . . 8 (𝐹𝑉 → dom 𝐹 ∈ V)
543ad2ant2 1131 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → dom 𝐹 ∈ V)
6 simp3 1135 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 0𝑊)
7 elsuppfn 7825 . . . . . . 7 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ V ∧ 0𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 )))
83, 5, 6, 7syl3anc 1368 . . . . . 6 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 )))
98anbi1d 632 . . . . 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 472 . . . . . 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 251 . . . . . . . . . . 11 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 )))
1312impl 459 . . . . . . . . . 10 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → 𝑥 ∈ (𝐹 supp 0 ))
1413fvresd 6669 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
1514eqeq1d 2803 . . . . . . . 8 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝑦))
1615pm5.32da 582 . . . . . . 7 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦)))
17 ancom 464 . . . . . . . 8 (((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦 ∧ (𝐹𝑥) ≠ 0 ))
18 simpr 488 . . . . . . . . . 10 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) = 𝑦)
1918neeq1d 3049 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ≠ 0𝑦0 ))
2019pm5.32da 582 . . . . . . . 8 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) = 𝑦 ∧ (𝐹𝑥) ≠ 0 ) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2117, 20syl5bb 286 . . . . . . 7 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2216, 21bitrd 282 . . . . . 6 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2322pm5.32da 582 . . . . 5 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) = 𝑦𝑦0 ))))
249, 11, 233bitrd 308 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ (𝐹 supp 0 ) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) = 𝑦𝑦0 ))))
2524rexbidv2 3257 . . 3 ((Fun 𝐹𝐹𝑉0𝑊) → (∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 )))
26 suppssdm 7830 . . . . 5 (𝐹 supp 0 ) ⊆ dom 𝐹
27 fnssres 6446 . . . . 5 ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
283, 26, 27sylancl 589 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29 fvelrnb 6705 . . . 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 6705 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
3231anbi1d 632 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ ran 𝐹𝑦0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦𝑦0 )))
33 eldifsn 4683 . . . . 5 (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹𝑦0 ))
34 r19.41v 3303 . . . . 5 (∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 ) ↔ (∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦𝑦0 ))
3532, 33, 343bitr4g 317 . . . 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 314 . 2 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
3837eqrdv 2799 1 ((Fun 𝐹𝐹𝑉0𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∃wrex 3110  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  {csn 4528  dom cdm 5523  ran crn 5524   ↾ cres 5525  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328  (class class class)co 7139   supp csupp 7817 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818 This theorem is referenced by:  fsupprnfi  30455
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