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Theorem fsuppssindlem1 39541
 Description: Lemma for fsuppssind 39543. 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 6715 . 2 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
3 fvres 6671 . . . . 5 (𝑥𝑆 → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
43adantl 485 . . . 4 (((𝜑𝑥𝐼) ∧ 𝑥𝑆) → ((𝐹𝑆)‘𝑥) = (𝐹𝑥))
5 eldif 3892 . . . . . 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 7858 . . . . . . 7 ((𝜑𝑥 ∈ (𝐼𝑆)) → (𝐹𝑥) = 0 )
109eqcomd 2804 . . . . . 6 ((𝜑𝑥 ∈ (𝐼𝑆)) → 0 = (𝐹𝑥))
115, 10sylan2br 597 . . . . 5 ((𝜑 ∧ (𝑥𝐼 ∧ ¬ 𝑥𝑆)) → 0 = (𝐹𝑥))
1211anassrs 471 . . . 4 (((𝜑𝑥𝐼) ∧ ¬ 𝑥𝑆) → 0 = (𝐹𝑥))
134, 12ifeqda 4462 . . 3 ((𝜑𝑥𝐼) → if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 ) = (𝐹𝑥))
1413mpteq2dva 5128 . 2 (𝜑 → (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )) = (𝑥𝐼 ↦ (𝐹𝑥)))
152, 14eqtr4d 2836 1 (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3879   ⊆ wss 3882  ifcif 4427   ↦ cmpt 5113   ↾ cres 5524  ⟶wf 6325  ‘cfv 6329  (class class class)co 7142   supp csupp 7823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-supp 7824 This theorem is referenced by:  fsuppssind  39543
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