Theorem fsuppssindlem2 42568

Description: Lemma for fsuppssind 42569. 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 6825	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
2	1	ifeq1d 4498	. . . . 5 ⊢ (𝑓 = 𝐹 → if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 ) = if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 ))
3	2	mpteq2dv 5189	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )))
4	3	eleq1d 2813	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻))
5	4	elrab 3650	. 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 5266	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
10	6, 9	elmapd 8774	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝑆) ↔ 𝐹:𝑆⟶𝐵))
11	10	anbi1d 631	. . 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 6633	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 ))
13		sseqin2 4176	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑆) = 𝑆)
14	8, 13	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∩ 𝑆) = 𝑆)
15	14	mpteq1d 5185	. . . . . . . . 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 6895	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → 𝐹 = (𝑥 ∈ 𝑆 ↦ (𝐹‘𝑥)))
19	16, 18	eqtr4d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) = 𝐹)
20		fconstmpt 5685	. . . . . . . . 9 ⊢ ((𝐼 ∖ 𝑆) × { 0 }) = (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )
21	20	eqcomi 2738	. . . . . . . 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 4122	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
24	12, 23	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
25	24	eleq1d 2813	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻 ↔ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))
26	25	pm5.32da 579	. . 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 279	. 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	bitrid 283	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396  Vcvv 3438   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ifcif 4478  {csn 4579   ↦ cmpt 5176   × cxp 5621  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   ↑_m cmap 8760

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-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762

This theorem is referenced by:  fsuppssind  42569