Theorem restuni4 39304

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	|- ( ph -> A e. V )
restuni4.2	|- ( ph -> B C_ U. A )

Assertion

Ref	Expression
restuni4	|- ( ph -> U. ( A ↾t B ) = B )

Proof of Theorem restuni4

Step	Hyp	Ref	Expression
1		incom 3805	. . 3 |- ( B i^i U. A ) = ( U. A i^i B )
2	1	a1i 11	. 2 |- ( ph -> ( B i^i U. A ) = ( U. A i^i B ) )
3		restuni4.2	. . 3 |- ( ph -> B C_ U. A )
4		dfss 3589	. . 3 |- ( B C_ U. A <-> B = ( B i^i U. A ) )
5	3, 4	sylib 208	. 2 |- ( ph -> B = ( B i^i U. A ) )
6		restuni4.1	. . 3 |- ( ph -> A e. V )
7	6	uniexd 39281	. . . 4 |- ( ph -> U. A e. _V )
8	7, 3	ssexd 4805	. . 3 |- ( ph -> B e. _V )
9	6, 8	restuni3 39301	. 2 |- ( ph -> U. ( A ↾t B ) = ( U. A i^i B ) )
10	2, 5, 9	3eqtr4rd 2667	1 |- ( ph -> U. ( A ↾t B ) = B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436  (class class class)co 6650   ↾_t crest 16081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by:  restuni6  39305  restuni5  39306  subsaluni  40578  issmflelem  40953  smfpimltxr  40956  issmfgtlem  40964  issmfgt  40965  issmfgelem  40977  smfpimgtxr  40988  smfresal  40995