Theorem restsubel 46441

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 4113	. . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴))
3	2	eqeq2d 2774	. . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
4	3	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
5		incom 4109	. . . . . 6 ⊢ (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽))
7		restsubel.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
8		dfss2 4097	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
9	7, 8	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴)
10	6, 9	eqtrd 2798	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴)
11	10	eqcomd 2769	. . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴))
12	1, 4, 11	rspcedvd 3582	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))
13		restsubel.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
14	1, 7	ssexd 5293	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
15		elrest 17648	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
16	13, 14, 15	syl2anc 596	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
17	12, 16	mpbird 260	1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4872  (class class class)co 7451   ↾_t crest 17641

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-rest 17643

This theorem is used by:  toprestsubel  46442