Theorem fsuppssindlem1 40940

Description: Lemma for fsuppssind 40942. 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 6945	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
3		fvres 6896	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
4	3	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
5		eldif 3953	. . . . . 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 8162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 )
10	9	eqcomd 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → 0 = (𝐹‘𝑥))
11	5, 10	sylan2br 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝑆)) → 0 = (𝐹‘𝑥))
12	11	anassrs 468	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝑆) → 0 = (𝐹‘𝑥))
13	4, 12	ifeqda 4557	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥))
14	13	mpteq2dva 5240	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
15	2, 14	eqtr4d 2774	1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∖ cdif 3940   ⊆ wss 3943  ifcif 4521   ↦ cmpt 5223   ↾ cres 5670  ⟶wf 6527  ‘cfv 6531  (class class class)co 7392   supp csupp 8127

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5277  ax-sep 5291  ax-nul 5298  ax-pr 5419  ax-un 7707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3474  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4991  df-br 5141  df-opab 5203  df-mpt 5224  df-id 5566  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-iota 6483  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7395  df-oprab 7396  df-mpo 7397  df-supp 8128

This theorem is referenced by:  fsuppssind  40942