Theorem restuni5 46411

Description: The underlying set of a subspace induced by the ↾_t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
restuni5.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
restuni5	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))

Proof of Theorem restuni5

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		id 23	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		restuni5.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	sseqtrdi 3953	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ ∪ 𝐽)
5	4	adantl 487	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽)
6	1, 5	restuni4 46409	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) = 𝐴)
7	6	eqcomd 2769	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∪ 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: (None)