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Theorem fsuppssindlem1 42601
Description: Lemma for fsuppssind 42603. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
Hypotheses
Ref Expression
fsuppssindlem1.z (𝜑0𝑊)
fsuppssindlem1.v (𝜑𝐼𝑉)
fsuppssindlem1.1 (𝜑𝐹:𝐼𝐵)
fsuppssindlem1.2 (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)
Assertion
Ref Expression
fsuppssindlem1 (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐹   𝑥,𝐼
Allowed substitution hints:   𝐵(𝑥)   𝑆(𝑥)   𝑉(𝑥)   𝑊(𝑥)   0 (𝑥)

Proof of Theorem fsuppssindlem1
StepHypRef Expression
1 fsuppssindlem1.1 . . 3 (𝜑𝐹:𝐼𝐵)
21feqmptd 6977 . 2 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
3 fvres 6925 . . . . 5 (𝑥𝑆 → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
43adantl 481 . . . 4 (((𝜑𝑥𝐼) ∧ 𝑥𝑆) → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
5 eldif 3961 . . . . . 6 (𝑥 ∈ (𝐼𝑆) ↔ (𝑥𝐼 ∧ ¬ 𝑥𝑆))
6 fsuppssindlem1.2 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)
7 fsuppssindlem1.v . . . . . . . 8 (𝜑𝐼𝑉)
8 fsuppssindlem1.z . . . . . . . 8 (𝜑0𝑊)
91, 6, 7, 8suppssr 8220 . . . . . . 7 ((𝜑𝑥 ∈ (𝐼𝑆)) → (𝐹𝑥) = 0 )
109eqcomd 2743 . . . . . 6 ((𝜑𝑥 ∈ (𝐼𝑆)) → 0 = (𝐹𝑥))
115, 10sylan2br 595 . . . . 5 ((𝜑 ∧ (𝑥𝐼 ∧ ¬ 𝑥𝑆)) → 0 = (𝐹𝑥))
1211anassrs 467 . . . 4 (((𝜑𝑥𝐼) ∧ ¬ 𝑥𝑆) → 0 = (𝐹𝑥))
134, 12ifeqda 4562 . . 3 ((𝜑𝑥𝐼) → if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 ) = (𝐹𝑥))
1413mpteq2dva 5242 . 2 (𝜑 → (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )) = (𝑥𝐼 ↦ (𝐹𝑥)))
152, 14eqtr4d 2780 1 (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  wss 3951  ifcif 4525  cmpt 5225  cres 5687  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-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:  fsuppssind  42603
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