Theorem restsubel 44150

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 4206	. . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴))
3	2	eqeq2d 2742	. . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
4	3	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
5		incom 4202	. . . . . 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 3966	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
9	7, 8	sylib 217	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴)
10	6, 9	eqtrd 2771	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴)
11	10	eqcomd 2737	. . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴))
12	1, 4, 11	rspcedvd 3615	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))
13		restsubel.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
14	1, 7	ssexd 5325	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
15		elrest 17378	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
16	13, 14, 15	syl2anc 583	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
17	12, 16	mpbird 256	1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473   ∩ cin 3948   ⊆ wss 3949  ∪ cuni 4909  (class class class)co 7412   ↾_t crest 17371

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rest 17373

This theorem is referenced by:  toprestsubel  44151