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Theorem fsuppssindlem1 42577
Description: Lemma for fsuppssind 42579. 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 6976 . 2 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
3 fvres 6925 . . . . 5 (𝑥𝑆 → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
43adantl 481 . . . 4 (((𝜑𝑥𝐼) ∧ 𝑥𝑆) → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
5 eldif 3972 . . . . . 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 8218 . . . . . . 7 ((𝜑𝑥 ∈ (𝐼𝑆)) → (𝐹𝑥) = 0 )
109eqcomd 2740 . . . . . 6 ((𝜑𝑥 ∈ (𝐼𝑆)) → 0 = (𝐹𝑥))
115, 10sylan2br 595 . . . . 5 ((𝜑 ∧ (𝑥𝐼 ∧ ¬ 𝑥𝑆)) → 0 = (𝐹𝑥))
1211anassrs 467 . . . 4 (((𝜑𝑥𝐼) ∧ ¬ 𝑥𝑆) → 0 = (𝐹𝑥))
134, 12ifeqda 4566 . . 3 ((𝜑𝑥𝐼) → if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 ) = (𝐹𝑥))
1413mpteq2dva 5247 . 2 (𝜑 → (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )) = (𝑥𝐼 ↦ (𝐹𝑥)))
152, 14eqtr4d 2777 1 (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  cdif 3959  wss 3962  ifcif 4530  cmpt 5230  cres 5690  wf 6558  cfv 6562  (class class class)co 7430   supp csupp 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184
This theorem is referenced by:  fsuppssind  42579
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