Theorem fsuppssindlem2 40204

Description: Lemma for fsuppssind 40205. Write a function as a union. (Contributed by SN, 15-Jul-2024.)

Hypotheses

Ref	Expression
fsuppssindlem2.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsuppssindlem2.v	⊢ (𝜑 → 𝐼 ∈ 𝑉)
fsuppssindlem2.s	⊢ (𝜑 → 𝑆 ⊆ 𝐼)

Assertion

Ref	Expression
fsuppssindlem2	⊢ (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))

Distinct variable groups:   𝑓,𝐼,𝑥   𝑆,𝑓,𝑥   𝑓,𝐹,𝑥   0 ,𝑓,𝑥   𝑓,𝐻   𝐵,𝑓

Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥)   𝐻(𝑥)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)

Proof of Theorem fsuppssindlem2

Step	Hyp	Ref	Expression
1		fveq1 6755	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
2	1	ifeq1d 4475	. . . . 5 ⊢ (𝑓 = 𝐹 → if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 ) = if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 ))
3	2	mpteq2dv 5172	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )))
4	3	eleq1d 2823	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻))
5	4	elrab 3617	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹 ∈ (𝐵 ↑m 𝑆) ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻))
6		fsuppssindlem2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		fsuppssindlem2.v	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8		fsuppssindlem2.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐼)
9	7, 8	ssexd 5243	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
10	6, 9	elmapd 8587	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝑆) ↔ 𝐹:𝑆⟶𝐵))
11	10	anbi1d 629	. . 3 ⊢ (𝜑 → ((𝐹 ∈ (𝐵 ↑m 𝑆) ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆⟶𝐵 ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻)))
12		partfun 6564	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 ))
13		sseqin2 4146	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑆) = 𝑆)
14	8, 13	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∩ 𝑆) = 𝑆)
15	14	mpteq1d 5165	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑆 ↦ (𝐹‘𝑥)))
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝑆 ↦ (𝐹‘𝑥)))
17		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → 𝐹:𝑆⟶𝐵)
18	17	feqmptd 6819	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → 𝐹 = (𝑥 ∈ 𝑆 ↦ (𝐹‘𝑥)))
19	16, 18	eqtr4d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) = 𝐹)
20		fconstmpt 5640	. . . . . . . . 9 ⊢ ((𝐼 ∖ 𝑆) × { 0 }) = (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )
21	20	eqcomi 2747	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 ) = ((𝐼 ∖ 𝑆) × { 0 })
22	21	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 ) = ((𝐼 ∖ 𝑆) × { 0 }))
23	19, 22	uneq12d 4094	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
24	12, 23	syl5eq 2791	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
25	24	eleq1d 2823	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻 ↔ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))
26	25	pm5.32da 578	. . 3 ⊢ (𝜑 → ((𝐹:𝑆⟶𝐵 ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
27	11, 26	bitrd 278	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝐵 ↑m 𝑆) ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
28	5, 27	syl5bb 282	1 ⊢ (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ifcif 4456  {csn 4558   ↦ cmpt 5153   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575

This theorem is referenced by:  fsuppssind  40205