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Theorem ressupprn 32176
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 6579 . . . . . . . . 9 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 215 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
323ad2ant1 1132 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 𝐹 Fn dom 𝐹)
4 dmexg 7897 . . . . . . . 8 (𝐹𝑉 → dom 𝐹 ∈ V)
543ad2ant2 1133 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → dom 𝐹 ∈ V)
6 simp3 1137 . . . . . . 7 ((Fun 𝐹𝐹𝑉0𝑊) → 0𝑊)
7 elsuppfn 8159 . . . . . . 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 629 . . . . 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 247 . . . . . . . . . . 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 6912 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹𝑥))
1514eqeq1d 2733 . . . . . . . 8 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) ≠ 0 ) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝑦))
1615pm5.32da 578 . . . . . . 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 2999 . . . . . . . . 9 ((((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ≠ 0𝑦0 ))
2019pm5.32da 578 . . . . . . . 8 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) = 𝑦 ∧ (𝐹𝑥) ≠ 0 ) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2117, 20bitrid 282 . . . . . . 7 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ (𝐹𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2216, 21bitrd 278 . . . . . 6 (((Fun 𝐹𝐹𝑉0𝑊) ∧ 𝑥 ∈ dom 𝐹) → (((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦) ↔ ((𝐹𝑥) = 𝑦𝑦0 )))
2322pm5.32da 578 . . . . 5 ((Fun 𝐹𝐹𝑉0𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ≠ 0 ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = 𝑦)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) = 𝑦𝑦0 ))))
249, 11, 233bitrd 304 . . . 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 8165 . . . . 5 (𝐹 supp 0 ) ⊆ dom 𝐹
27 fnssres 6674 . . . . 5 ((𝐹 Fn dom 𝐹 ∧ (𝐹 supp 0 ) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
283, 26, 27sylancl 585 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → (𝐹 ↾ (𝐹 supp 0 )) Fn (𝐹 supp 0 ))
29 fvelrnb 6953 . . . 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 6953 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
3231anbi1d 629 . . . . 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 313 . . . 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 310 . 2 ((Fun 𝐹𝐹𝑉0𝑊) → (𝑦 ∈ ran (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑦 ∈ (ran 𝐹 ∖ { 0 })))
3837eqrdv 2729 1 ((Fun 𝐹𝐹𝑉0𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069  Vcvv 3473  cdif 3946  wss 3949  {csn 4629  dom cdm 5677  ran crn 5678  cres 5679  Fun wfun 6538   Fn wfn 6539  cfv 6544  (class class class)co 7412   supp csupp 8149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8150
This theorem is referenced by:  fsupprnfi  32178
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