Theorem fsuppssindlem1 42949

Description: Lemma for fsuppssind 42951. 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 6910	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
3		fvres 6861	. . . . 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 8147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → (𝐹‘𝑥) = 0 )
10	9	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑆)) → 0 = (𝐹‘𝑥))
11	5, 10	sylan2br 596	. . . . 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 4518	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 ) = (𝐹‘𝑥))
14	13	mpteq2dva 5193	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
15	2, 14	eqtr4d 2775	1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ⊆ wss 3903  ifcif 4481   ↦ cmpt 5181   ↾ cres 5634  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   supp csupp 8112

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-supp 8113

This theorem is referenced by:  fsuppssind  42951