Theorem fsuppssindlem1 42564

Description: Lemma for fsuppssind 42566. 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 6891	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
3		fvres 6841	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
4	3	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ↾ 𝑆)‘𝑥) = (𝐹‘𝑥))
5		eldif 3913	. . . . . 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 8128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 )
10	9	eqcomd 2735	. . . . . 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 467	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝑆) → 0 = (𝐹‘𝑥))
13	4, 12	ifeqda 4513	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥))
14	13	mpteq2dva 5185	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
15	2, 14	eqtr4d 2767	1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900   ⊆ wss 3903  ifcif 4476   ↦ cmpt 5173   ↾ cres 5621  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   supp csupp 8093

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-supp 8094

This theorem is referenced by:  fsuppssind  42566