Theorem recexprlemell 7423

Description: Membership in the lower cut of B. Lemma for recexpr 7439. (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
recexprlemell	|- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

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

Proof of Theorem recexprlemell

Step	Hyp	Ref	Expression
1		elex 2692	. 2 |- ( C e. ( 1st ` B ) -> C e. _V )
2		ltrelnq 7166	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4586	. . . . . 6 |- ( C <Q y -> ( C e. Q. /\ y e. Q. ) )
4	3	simpld 111	. . . . 5 |- ( C <Q y -> C e. Q. )
5		elex 2692	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( C <Q y -> C e. _V )
7	6	adantr 274	. . 3 |- ( ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
8	7	exlimiv 1577	. 2 |- ( E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
9		breq1 3927	. . . . 5 |- ( x = C -> ( x <Q y <-> C <Q y ) )
10	9	anbi1d 460	. . . 4 |- ( x = C -> ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
11	10	exbidv 1797	. . 3 |- ( x = C -> ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` 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 5417	. . . 4 |- ( 1st ` B ) = ( 1st ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. )
14		nqex 7164	. . . . . 6 |- Q. e. _V
15	2	brel 4586	. . . . . . . . . 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 1577	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3167	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4061	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4586	. . . . . . . . . 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 1577	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3167	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4061	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op1st 6037	. . . 4 |- ( 1st ` <. { 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 ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
28	13, 27	eqtri 2158	. . 3 |- ( 1st ` B ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
29	11, 28	elab2g 2826	. 2 |- ( C e. _V -> ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
30	1, 8, 29	pm5.21nii 693	1 |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   _Vcvv 2681   <.cop 3525   class class class wbr 3924   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   *Qcrq 7085   <Q cltq 7086

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-qs 6428  df-ni 7105  df-nqqs 7149  df-ltnqqs 7154

This theorem is referenced by:  recexprlemm  7425  recexprlemopl  7426  recexprlemlol  7427  recexprlemdisj  7431  recexprlemloc  7432  recexprlem1ssl  7434  recexprlemss1l  7436