Theorem clrellem 37929

Description: When the property ps holds for a relation substituted for x, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)

Hypotheses

Ref	Expression
clrellem.y	|- ( ph -> Y e. _V )
clrellem.rel	|- ( ph -> Rel X )
clrellem.sub	|- ( x = `' `' Y -> ( ps <-> ch ) )
clrellem.sup	|- ( ph -> X C_ Y )
clrellem.maj	|- ( ph -> ch )

Assertion

Ref	Expression
clrellem	|- ( ph -> Rel |^| { x | ( X C_ x /\ ps ) } )

Distinct variable groups:   x, X   x, Y   ph, x   ch, x

Allowed substitution hint:   ps( x)

Proof of Theorem clrellem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clrellem.y	. . . 4 |- ( ph -> Y e. _V )
2		cnvexg 7112	. . . 4 |- ( Y e. _V -> `' Y e. _V )
3		cnvexg 7112	. . . 4 |- ( `' Y e. _V -> `' `' Y e. _V )
4	1, 2, 3	3syl 18	. . 3 |- ( ph -> `' `' Y e. _V )
5		clrellem.rel	. . . . . 6 |- ( ph -> Rel X )
6		dfrel2 5583	. . . . . 6 |- ( Rel X <-> `' `' X = X )
7	5, 6	sylib 208	. . . . 5 |- ( ph -> `' `' X = X )
8		clrellem.sup	. . . . . 6 |- ( ph -> X C_ Y )
9		cnvss 5294	. . . . . 6 |- ( X C_ Y -> `' X C_ `' Y )
10		cnvss 5294	. . . . . 6 |- ( `' X C_ `' Y -> `' `' X C_ `' `' Y )
11	8, 9, 10	3syl 18	. . . . 5 |- ( ph -> `' `' X C_ `' `' Y )
12	7, 11	eqsstr3d 3640	. . . 4 |- ( ph -> X C_ `' `' Y )
13		clrellem.maj	. . . 4 |- ( ph -> ch )
14		relcnv 5503	. . . . 5 |- Rel `' `' Y
15	14	a1i 11	. . . 4 |- ( ph -> Rel `' `' Y )
16	12, 13, 15	jca31 557	. . 3 |- ( ph -> ( ( X C_ `' `' Y /\ ch ) /\ Rel `' `' Y ) )
17		clrellem.sub	. . . . 5 |- ( x = `' `' Y -> ( ps <-> ch ) )
18	17	cleq2lem 37914	. . . 4 |- ( x = `' `' Y -> ( ( X C_ x /\ ps ) <-> ( X C_ `' `' Y /\ ch ) ) )
19		releq 5201	. . . 4 |- ( x = `' `' Y -> ( Rel x <-> Rel `' `' Y ) )
20	18, 19	anbi12d 747	. . 3 |- ( x = `' `' Y -> ( ( ( X C_ x /\ ps ) /\ Rel x ) <-> ( ( X C_ `' `' Y /\ ch ) /\ Rel `' `' Y ) ) )
21	4, 16, 20	elabd 3352	. 2 |- ( ph -> E. x ( ( X C_ x /\ ps ) /\ Rel x ) )
22		releq 5201	. . . 4 |- ( y = x -> ( Rel y <-> Rel x ) )
23	22	rexab2 3373	. . 3 |- ( E. y e. { x | ( X C_ x /\ ps ) } Rel y <-> E. x ( ( X C_ x /\ ps ) /\ Rel x ) )
24	23	biimpri 218	. 2 |- ( E. x ( ( X C_ x /\ ps ) /\ Rel x ) -> E. y e. { x | ( X C_ x /\ ps ) } Rel y )
25		relint 5242	. 2 |- ( E. y e. { x | ( X C_ x /\ ps ) } Rel y -> Rel |^| { x | ( X C_ x /\ ps ) } )
26	21, 24, 25	3syl 18	1 |- ( ph -> Rel |^| { x | ( X C_ x /\ ps ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |^|cint 4475   `'ccnv 5113   Rel wrel 5119

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-iin 4523  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)