Theorem restsubel 45691

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 4163	. . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴))
3	2	eqeq2d 2772	. . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
4	3	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴)))
5		incom 4159	. . . . . 6 ⊢ (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽))
7		restsubel.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
8		dfss2 3920	. . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴)
9	7, 8	sylib 220	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴)
10	6, 9	eqtrd 2796	. . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴)
11	10	eqcomd 2767	. . 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 5277	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
15		elrest 17446	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
16	13, 14, 15	syl2anc 593	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)))
17	12, 16	mpbird 259	1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4862  (class class class)co 7390   ↾_t crest 17439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-rest 17441

This theorem is referenced by:  toprestsubel  45692