Theorem recexprlemelu 7624

Description: Membership in the upper cut of B. Lemma for recexpr 7639. (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 2750	. 2 |- ( C e. ( 2nd ` B ) -> C e. _V )
2		ltrelnq 7366	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4680	. . . . . 6 |- ( y <Q C -> ( y e. Q. /\ C e. Q. ) )
4	3	simprd 114	. . . . 5 |- ( y <Q C -> C e. Q. )
5		elex 2750	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( y <Q C -> C e. _V )
7	6	adantr 276	. . 3 |- ( ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
8	7	exlimiv 1598	. 2 |- ( E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
9		breq2 4009	. . . . 5 |- ( x = C -> ( y <Q x <-> y <Q C ) )
10	9	anbi1d 465	. . . 4 |- ( x = C -> ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
11	10	exbidv 1825	. . 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 5520	. . . 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 7364	. . . . . 6 |- Q. e. _V
15	2	brel 4680	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
16	15	simpld 112	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
17	16	adantr 276	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
18	17	exlimiv 1598	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3232	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4143	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4680	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
22	21	simprd 114	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
23	22	adantr 276	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
24	23	exlimiv 1598	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3232	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4143	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op2nd 6150	. . . 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 2198	. . 3 |- ( 2nd ` B ) = { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) }
29	11, 28	elab2g 2886	. 2 |- ( C e. _V -> ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
30	1, 8, 29	pm5.21nii 704	1 |- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   _Vcvv 2739   <.cop 3597   class class class wbr 4005   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   *Qcrq 7285   <Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-ltnqqs 7354

This theorem is referenced by:  recexprlemm  7625  recexprlemopu  7628  recexprlemupu  7629  recexprlemdisj  7631  recexprlemloc  7632  recexprlem1ssu  7635  recexprlemss1u  7637