Theorem restsubel 42745

Description: A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)

Hypotheses

Ref	Expression
restsubel.1	⊢ (𝜑 → 𝐽 ∈ 𝑉)
restsubel.2	⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)
restsubel.3	⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)

Assertion

Ref	Expression
restsubel	⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))

Proof of Theorem restsubel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		restsubel.2	. . 3 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)
2		ineq1 4145	. . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴))
3	2	eqeq2d 2747	. . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
4	3	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
5		incom 4141	. . . . . 6 ⊢ (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽))
7		restsubel.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
8		df-ss 3909	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
9	7, 8	sylib 217	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴)
10	6, 9	eqtrd 2776	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴)
11	10	eqcomd 2742	. . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴))
12	1, 4, 11	rspcedvd 3568	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))
13		restsubel.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
14	1, 7	ssexd 5257	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
15		elrest 17183	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
16	13, 14, 15	syl2anc 585	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
17	12, 16	mpbird 257	1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  ∃wrex 3071  Vcvv 3437   ∩ cin 3891   ⊆ wss 3892  ∪ cuni 4844  (class class class)co 7307   ↾_t crest 17176

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-rest 17178

This theorem is referenced by:  toprestsubel  42746