Theorem umgrislfupgrenlem 15806

Description: Lemma for umgrislfupgrdom 15807. (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 3449	. 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 6982	. . . . . . 7 |- -. 2o ~<_ 1o
3		domentr 6901	. . . . . . . 8 |- ( ( 2o ~<_ x /\ x ~~ 1o ) -> 2o ~<_ 1o )
4	3	ex 115	. . . . . . 7 |- ( 2o ~<_ x -> ( x ~~ 1o -> 2o ~<_ 1o ) )
5	2, 4	mtoi 666	. . . . . 6 |- ( 2o ~<_ x -> -. x ~~ 1o )
6		orel1 727	. . . . . 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 713	. . . . 5 |- ( x ~~ 2o -> ( x ~~ 1o \/ x ~~ 2o ) )
10		ensymb 6890	. . . . . 6 |- ( 2o ~~ x <-> x ~~ 2o )
11		endom 6872	. . . . . 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 2759	. 2 |- { x e. ~P V | ( ( x ~~ 1o \/ x ~~ 2o ) /\ 2o ~<_ x ) } = { x e. ~P V | x ~~ 2o }
16	1, 15	eqtri 2227	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 710   = wceq 1373   {crab 2489   i^i cin 3169   ~Pcpw 3621   class class class wbr 4054   1oc1o 6513   2oc2o 6514   ~~ cen 6843   ~<_ cdom 6844

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1o 6520  df-2o 6521  df-er 6638  df-en 6846  df-dom 6847

This theorem is referenced by:  umgrislfupgrdom  15807