Theorem restuni4 42623

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.)

Hypotheses

Ref	Expression
restuni4.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
restuni4.2	⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)

Assertion

Ref	Expression
restuni4	⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Proof of Theorem restuni4

Step	Hyp	Ref	Expression
1		incom 4139	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
3		restuni4.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)
4		dfss 3909	. . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴))
5	3, 4	sylib 217	. 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴))
6		restuni4.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	uniexd 7586	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
8	7, 3	ssexd 5251	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9	6, 8	restuni3 42620	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
10	2, 5, 9	3eqtr4rd 2790	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ∩ cin 3890   ⊆ wss 3891  ∪ cuni 4844  (class class class)co 7268   ↾_t crest 17112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-rest 17114

This theorem is referenced by:  restuni6  42624  restuni5  42625  subsaluni  43853  issmflelem  44231  smfpimltxr  44234  issmfgtlem  44242  issmfgt  44243  issmfgelem  44255  smfpimgtxr  44266  smfresal  44273