Theorem restuni5 40235

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 476	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ 𝑉)
2		id 22	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		restuni5.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	syl6sseq 3870	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ ∪ 𝐽)
5	4	adantl 475	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽)
6	1, 5	restuni4 40233	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) = 𝐴)
7	6	eqcomd 2784	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ⊆ wss 3792  ∪ cuni 4671  (class class class)co 6922   ↾_t crest 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-rest 16469

This theorem is referenced by: (None)