Theorem fsuppssindlem2 42625

Description: Lemma for fsuppssind 42626. 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 6816	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
2	1	ifeq1d 4490	. . . . 5 ⊢ (𝑓 = 𝐹 → if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 ) = if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 ))
3	2	mpteq2dv 5180	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )))
4	3	eleq1d 2816	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) ∈ 𝐻))
5	4	elrab 3642	. 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 5257	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ V)
10	6, 9	elmapd 8759	. . . 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 6623	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 ))
13		sseqin2 4168	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑆) = 𝑆)
14	8, 13	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∩ 𝑆) = 𝑆)
15	14	mpteq1d 5176	. . . . . . . . 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 6885	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → 𝐹 = (𝑥 ∈ 𝑆 ↦ (𝐹‘𝑥)))
19	16, 18	eqtr4d 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) = 𝐹)
20		fconstmpt 5673	. . . . . . . . 9 ⊢ ((𝐼 ∖ 𝑆) × { 0 }) = (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )
21	20	eqcomi 2740	. . . . . . . 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 4114	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → ((𝑥 ∈ (𝐼 ∩ 𝑆) ↦ (𝐹‘𝑥)) ∪ (𝑥 ∈ (𝐼 ∖ 𝑆) ↦ 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
24	12, 23	eqtrid 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑆⟶𝐵) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝐹‘𝑥), 0 )) = (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })))
25	24	eleq1d 2816	. . . 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 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ifcif 4470  {csn 4571   ↦ cmpt 5167   × cxp 5609  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-map 8747

This theorem is referenced by:  fsuppssind  42626