Theorem umgrislfupgrenlem 15980

Description: Lemma for umgrislfupgrdom 15981. (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 3479	. 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 7050	. . . . . . 7 |- -. 2o ~<_ 1o
3		domentr 6964	. . . . . . . 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 732	. . . . . 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 718	. . . . 5 |- ( x ~~ 2o -> ( x ~~ 1o \/ x ~~ 2o ) )
10		ensymb 6953	. . . . . 6 |- ( 2o ~~ x <-> x ~~ 2o )
11		endom 6935	. . . . . 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 2789	. 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 715   = wceq 1397   {crab 2514   i^i cin 3199   ~Pcpw 3652   class class class wbr 4088   1oc1o 6574   2oc2o 6575   ~~ cen 6906   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-dom 6910

This theorem is referenced by:  umgrislfupgrdom  15981  usgrislfuspgrdom  16040