Theorem umgrislfupgrenlem 16054

Description: Lemma for umgrislfupgrdom 16055. (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 3481	. 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 7094	. . . . . . 7 |- -. 2o ~<_ 1o
3		domentr 7008	. . . . . . . 8 |- ( ( 2o ~<_ x /\ x ~~ 1o ) -> 2o ~<_ 1o )
4	3	ex 115	. . . . . . 7 |- ( 2o ~<_ x -> ( x ~~ 1o -> 2o ~<_ 1o ) )
5	2, 4	mtoi 670	. . . . . 6 |- ( 2o ~<_ x -> -. x ~~ 1o )
6		orel1 733	. . . . . 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 719	. . . . 5 |- ( x ~~ 2o -> ( x ~~ 1o \/ x ~~ 2o ) )
10		ensymb 6997	. . . . . 6 |- ( 2o ~~ x <-> x ~~ 2o )
11		endom 6979	. . . . . 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 2790	. 2 |- { x e. ~P V | ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) } = { x e. ~P V | x ~~ 2o }
16	1, 15	eqtri 2252	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 716   = wceq 1398   {crab 2515   i^i cin 3200   ~Pcpw 3656   class class class wbr 4093   1oc1o 6618   2oc2o 6619   ~~ cen 6950   ~<_ cdom 6951

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-dom 6954

This theorem is referenced by:  umgrislfupgrdom  16055  usgrislfuspgrdom  16114