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Theorem ressupprn 32705
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 6598 . . . . . . . . 9 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 216 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
323ad2ant1 1132 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 𝐹 Fn dom 𝐹)
4 dmexg 7924 . . . . . . . 8 (𝐹𝑉 → dom 𝐹 ∈ V)
543ad2ant2 1133 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → dom 𝐹 ∈ V)
6 simp3 1137 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 0𝑊)
7 elsuppfn 8194 . . . . . . 7 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ V ∧ 0𝑊) → (𝑥 ∈ (𝐹 supp 0 ) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 0 )))
83, 5, 6, 7syl3anc 1370 . . . . . 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 6927 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
1514eqeq1d 2737 . . . . . . . 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 2998 . . . . . . . . 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 3173 . . 3 ((Fun 𝐹𝐹𝑉0𝑊) → (∃𝑥 ∈ (𝐹 supp 0 )((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹((𝐹𝑥) = 𝑦𝑦0 )))
26 suppssdm 8201 . . . . 5 (𝐹 supp 0 ) ⊆ dom 𝐹
27 fnssres 6692 . . . . 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 4791 . . . . 5 (𝑦 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑦 ∈ ran 𝐹𝑦0 ))
34 r19.41v 3187 . . . . 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 2733 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 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  {csn 4631  dom cdm 5689  ran crn 5690  cres 5691  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  fsupprnfi  32707
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