Theorem restsubel 45147

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 4176	. . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴))
3	2	eqeq2d 2740	. . . 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 4172	. . . . . 6 ⊢ (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽))
7		restsubel.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
8		dfss2 3932	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴)
10	6, 9	eqtrd 2764	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴)
11	10	eqcomd 2735	. . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴))
12	1, 4, 11	rspcedvd 3590	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))
13		restsubel.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
14	1, 7	ssexd 5279	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
15		elrest 17390	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
16	13, 14, 15	syl2anc 584	. 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 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  (class class class)co 7387   ↾_t crest 17383

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-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-rest 17385

This theorem is referenced by:  toprestsubel  45148