Theorem fsuppssindlem1 43400

Description: Lemma for fsuppssind 43402. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)

Hypotheses

Ref	Expression
fsuppssindlem1.z	⊢ (𝜑 → 0 ∈ 𝑊)
fsuppssindlem1.v	⊢ (𝜑 → 𝐼 ∈ 𝑉)
fsuppssindlem1.1	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
fsuppssindlem1.2	⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)

Assertion

Ref	Expression
fsuppssindlem1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘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

Step	Hyp	Ref	Expression
1		fsuppssindlem1.1	. . 3 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
2	1	feqmptd 6953	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
3		fvres 6904	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
4	3	adantl 487	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
5		eldif 3916	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑆) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆))
6		fsuppssindlem1.2	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)
7		fsuppssindlem1.v	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8		fsuppssindlem1.z	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝑊)
9	1, 6, 7, 8	suppssr 8197	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 )
10	9	eqcomd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → 0 = (𝐹‘𝑥))
11	5, 10	sylan2br 607	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆)) → 0 = (𝐹‘𝑥))
12	11	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝑆) → 0 = (𝐹‘𝑥))
13	4, 12	ifeqda 4526	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥))
14	13	mpteq2dva 5206	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
15	2, 14	eqtr4d 2803	1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ⊆ wss 3906  ifcif 4489   ↦ cmpt 5194   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   supp csupp 8162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-supp 8163

This theorem is used by:  fsuppssind  43402