Theorem recexprlemelu 7522

Description: Membership in the upper cut of B. Lemma for recexpr 7537. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemelu	|- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem recexprlemelu

Step	Hyp	Ref	Expression
1		elex 2720	. 2 |- ( C e. ( 2nd ` B ) -> C e. _V )
2		ltrelnq 7264	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4631	. . . . . 6 |- ( y <Q C -> ( y e. Q. /\ C e. Q. ) )
4	3	simprd 113	. . . . 5 |- ( y <Q C -> C e. Q. )
5		elex 2720	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( y <Q C -> C e. _V )
7	6	adantr 274	. . 3 |- ( ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
8	7	exlimiv 1575	. 2 |- ( E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
9		breq2 3965	. . . . 5 |- ( x = C -> ( y <Q x <-> y <Q C ) )
10	9	anbi1d 461	. . . 4 |- ( x = C -> ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
11	10	exbidv 1802	. . 3 |- ( x = C -> ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
12		recexpr.1	. . . . 5 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
13	12	fveq2i 5464	. . . 4 |- ( 2nd ` B ) = ( 2nd ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. )
14		nqex 7262	. . . . . 6 |- Q. e. _V
15	2	brel 4631	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
16	15	simpld 111	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
17	16	adantr 274	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
18	17	exlimiv 1575	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3199	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4098	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4631	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
22	21	simprd 113	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
23	22	adantr 274	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
24	23	exlimiv 1575	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3199	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4098	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op2nd 6085	. . . 4 |- ( 2nd ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. ) = { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) }
28	13, 27	eqtri 2175	. . 3 |- ( 2nd ` B ) = { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) }
29	11, 28	elab2g 2855	. 2 |- ( C e. _V -> ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
30	1, 8, 29	pm5.21nii 694	1 |- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 2125   {cab 2140   _Vcvv 2709   <.cop 3559   class class class wbr 3961   ` cfv 5163   1stc1st 6076   2ndc2nd 6077   Q.cnq 7179   *Qcrq 7183   <Q cltq 7184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-iinf 4541

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-2nd 6079  df-qs 6475  df-ni 7203  df-nqqs 7247  df-ltnqqs 7252

This theorem is referenced by:  recexprlemm  7523  recexprlemopu  7526  recexprlemupu  7527  recexprlemdisj  7529  recexprlemloc  7530  recexprlem1ssu  7533  recexprlemss1u  7535