Theorem restuni4 45698

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 4164	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
3		restuni4.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)
4		dfss 3926	. . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴))
5	3, 4	sylib 221	. 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴))
6		restuni4.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	uniexd 7729	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
8	7, 3	ssexd 5284	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9	6, 8	restuni3 45695	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
10	2, 5, 9	3eqtr4rd 2811	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4867  (class class class)co 7400   ↾_t crest 17461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-rest 17463

This theorem is referenced by:  restuni6  45699  restuni5  45700  subsaluni  46933  issmflelem  47317  issmfgtlem  47328  issmfgt  47329  issmfgelem  47342  smfresal  47361