Theorem salgenss 40554

Description: The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 40562, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
salgenss.x	|- ( ph -> X e. V )
salgenss.g	|- G = (SalGen ` X )
salgenss.s	|- ( ph -> S e. SAlg )
salgenss.i	|- ( ph -> X C_ S )
salgenss.u	|- ( ph -> U. S = U. X )

Assertion

Ref	Expression
salgenss	|- ( ph -> G C_ S )

Proof of Theorem salgenss

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		salgenss.g	. . . 4 |- G = (SalGen ` X )
2	1	a1i 11	. . 3 |- ( ph -> G = (SalGen ` X ) )
3		salgenss.x	. . . 4 |- ( ph -> X e. V )
4		salgenval 40541	. . . 4 |- ( X e. V -> (SalGen ` X ) = |^| { s e. SAlg | ( U. s = U. X /\ X C_ s ) } )
5	3, 4	syl 17	. . 3 |- ( ph -> (SalGen ` X ) = |^| { s e. SAlg | ( U. s = U. X /\ X C_ s ) } )
6	2, 5	eqtrd 2656	. 2 |- ( ph -> G = |^| { s e. SAlg | ( U. s = U. X /\ X C_ s ) } )
7		salgenss.s	. . . . 5 |- ( ph -> S e. SAlg )
8		salgenss.u	. . . . . 6 |- ( ph -> U. S = U. X )
9		salgenss.i	. . . . . 6 |- ( ph -> X C_ S )
10	8, 9	jca 554	. . . . 5 |- ( ph -> ( U. S = U. X /\ X C_ S ) )
11	7, 10	jca 554	. . . 4 |- ( ph -> ( S e. SAlg /\ ( U. S = U. X /\ X C_ S ) ) )
12		unieq 4444	. . . . . . 7 |- ( s = S -> U. s = U. S )
13	12	eqeq1d 2624	. . . . . 6 |- ( s = S -> ( U. s = U. X <-> U. S = U. X ) )
14		sseq2 3627	. . . . . 6 |- ( s = S -> ( X C_ s <-> X C_ S ) )
15	13, 14	anbi12d 747	. . . . 5 |- ( s = S -> ( ( U. s = U. X /\ X C_ s ) <-> ( U. S = U. X /\ X C_ S ) ) )
16	15	elrab 3363	. . . 4 |- ( S e. { s e. SAlg | ( U. s = U. X /\ X C_ s ) } <-> ( S e. SAlg /\ ( U. S = U. X /\ X C_ S ) ) )
17	11, 16	sylibr 224	. . 3 |- ( ph -> S e. { s e. SAlg | ( U. s = U. X /\ X C_ s ) } )
18		intss1 4492	. . 3 |- ( S e. { s e. SAlg | ( U. s = U. X /\ X C_ s ) } -> |^| { s e. SAlg | ( U. s = U. X /\ X C_ s ) } C_ S )
19	17, 18	syl 17	. 2 |- ( ph -> |^| { s e. SAlg | ( U. s = U. X /\ X C_ s ) } C_ S )
20	6, 19	eqsstrd 3639	1 |- ( ph -> G C_ S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   U.cuni 4436   |^|cint 4475   ` cfv 5888  SAlgcsalg 40528  SalGencsalgen 40532

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-sep 4781  ax-nul 4789  ax-pow 4843  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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  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-iota 5851  df-fun 5890  df-fv 5896  df-salg 40529  df-salgen 40533

This theorem is referenced by:  issalgend  40556  dfsalgen2  40559  borelmbl  40850  smfpimbor1lem2  41006