Theorem umgrislfupgrenlem 15969

Description: Lemma for umgrislfupgrdom 15970. (Contributed by AV, 27-Jan-2021.)

Assertion

Ref	Expression
umgrislfupgrenlem	|- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }

Proof of Theorem umgrislfupgrenlem

Step	Hyp	Ref	Expression
1		inrab 3477	. 2 |- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) }
2		1ndom2 7046	. . . . . . 7 |- -. 2o ~<_ 1o
3		domentr 6960	. . . . . . . 8 |- ( ( 2o ~<_ x /\ x ~~ 1o ) -> 2o ~<_ 1o )
4	3	ex 115	. . . . . . 7 |- ( 2o ~<_ x -> ( x ~~ 1o -> 2o ~<_ 1o ) )
5	2, 4	mtoi 668	. . . . . 6 |- ( 2o ~<_ x -> -. x ~~ 1o )
6		orel1 730	. . . . . 6 |- ( -. x ~~ 1o -> ( ( x ~~ 1o \/ x ~~ 2o ) -> x ~~ 2o ) )
7	5, 6	syl 14	. . . . 5 |- ( 2o ~<_ x -> ( ( x ~~ 1o \/ x ~~ 2o ) -> x ~~ 2o ) )
8	7	impcom 125	. . . 4 |- ( ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) -> x ~~ 2o )
9		olc 716	. . . . 5 |- ( x ~~ 2o -> ( x ~~ 1o \/ x ~~ 2o ) )
10		ensymb 6949	. . . . . 6 |- ( 2o ~~ x <-> x ~~ 2o )
11		endom 6931	. . . . . 6 |- ( 2o ~~ x -> 2o ~<_ x )
12	10, 11	sylbir 135	. . . . 5 |- ( x ~~ 2o -> 2o ~<_ x )
13	9, 12	jca 306	. . . 4 |- ( x ~~ 2o -> ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) )
14	8, 13	impbii 126	. . 3 |- ( ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) <-> x ~~ 2o )
15	14	rabbii 2787	. 2 |- { x e. ~P V | ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) } = { x e. ~P V | x ~~ 2o }
16	1, 15	eqtri 2250	1 |- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   {crab 2512   i^i cin 3197   ~Pcpw 3650   class class class wbr 4086   1oc1o 6570   2oc2o 6571   ~~ cen 6902   ~<_ cdom 6903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-dom 6906

This theorem is referenced by:  umgrislfupgrdom  15970  usgrislfuspgrdom  16029