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Theorem restuni5 41753
 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 ((𝐽𝑉𝐴𝑋) → 𝐴 = (𝐽t 𝐴))

Proof of Theorem restuni5
StepHypRef Expression
1 simpl 486 . . 3 ((𝐽𝑉𝐴𝑋) → 𝐽𝑉)
2 id 22 . . . . 5 (𝐴𝑋𝐴𝑋)
3 restuni5.1 . . . . 5 𝑋 = 𝐽
42, 3sseqtrdi 3965 . . . 4 (𝐴𝑋𝐴 𝐽)
54adantl 485 . . 3 ((𝐽𝑉𝐴𝑋) → 𝐴 𝐽)
61, 5restuni4 41751 . 2 ((𝐽𝑉𝐴𝑋) → (𝐽t 𝐴) = 𝐴)
76eqcomd 2804 1 ((𝐽𝑉𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∪ cuni 4800  (class class class)co 7135   ↾t crest 16686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-rest 16688 This theorem is referenced by: (None)
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