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Theorem suppssrg 8221
Description: A function is zero outside its support. Version of suppssr 8220 avoiding ax-rep 5279 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)
Hypotheses
Ref Expression
suppssrg.f (𝜑𝐹:𝐴𝐵)
suppssrg.n (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
suppssrg.a (𝜑𝐹𝑉)
suppssrg.z (𝜑𝑍𝑈)
Assertion
Ref Expression
suppssrg ((𝜑𝑋 ∈ (𝐴𝑊)) → (𝐹𝑋) = 𝑍)

Proof of Theorem suppssrg
StepHypRef Expression
1 eldif 3961 . 2 (𝑋 ∈ (𝐴𝑊) ↔ (𝑋𝐴 ∧ ¬ 𝑋𝑊))
2 suppssrg.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
32ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 suppssrg.a . . . . . . 7 (𝜑𝐹𝑉)
5 suppssrg.z . . . . . . 7 (𝜑𝑍𝑈)
6 elsuppfng 8194 . . . . . . 7 ((𝐹 Fn 𝐴𝐹𝑉𝑍𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋𝐴 ∧ (𝐹𝑋) ≠ 𝑍)))
73, 4, 5, 6syl3anc 1373 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋𝐴 ∧ (𝐹𝑋) ≠ 𝑍)))
8 suppssrg.n . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
98sseld 3982 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋𝑊))
107, 9sylbird 260 . . . . 5 (𝜑 → ((𝑋𝐴 ∧ (𝐹𝑋) ≠ 𝑍) → 𝑋𝑊))
1110expdimp 452 . . . 4 ((𝜑𝑋𝐴) → ((𝐹𝑋) ≠ 𝑍𝑋𝑊))
1211necon1bd 2958 . . 3 ((𝜑𝑋𝐴) → (¬ 𝑋𝑊 → (𝐹𝑋) = 𝑍))
1312impr 454 . 2 ((𝜑 ∧ (𝑋𝐴 ∧ ¬ 𝑋𝑊)) → (𝐹𝑋) = 𝑍)
141, 13sylan2b 594 1 ((𝜑𝑋 ∈ (𝐴𝑊)) → (𝐹𝑋) = 𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  cdif 3948  wss 3951   Fn wfn 6556  wf 6557  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-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-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  psrbaglesupp  21942  psrbaglefi  21946  mhpmulcl  22153  mhpvscacl  22158  evlsvvvallem2  42572  selvvvval  42595
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