Theorem umgrislfupgrenlem 15922

Description: Lemma for umgrislfupgrdom 15923. (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 3476	. 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 7022	. . . . . . 7 |- -. 2o ~<_ 1o
3		domentr 6941	. . . . . . . 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 6930	. . . . . 6 |- ( 2o ~~ x <-> x ~~ 2o )
11		endom 6912	. . . . . 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 2785	. 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 3196   ~Pcpw 3649   class class class wbr 4082   1oc1o 6553   2oc2o 6554   ~~ cen 6883   ~<_ cdom 6884

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886  df-dom 6887

This theorem is referenced by:  umgrislfupgrdom  15923  usgrislfuspgrdom  15982