Theorem recexprlemopl 7566

Description: The lower cut of B is open. Lemma for recexpr 7579. (Contributed by Jim Kingdon, 28-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
recexprlemopl	|- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemopl

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
2	1	recexprlemell 7563	. . 3 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
3		ltbtwnnqq 7356	. . . . . 6 |- ( q <Q y <-> E. r e. Q. ( q <Q r /\ r <Q y ) )
4	3	biimpi 119	. . . . 5 |- ( q <Q y -> E. r e. Q. ( q <Q r /\ r <Q y ) )
5		simpll 519	. . . . . . . 8 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> q <Q r )
6		19.8a 1578	. . . . . . . . . 10 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
7	1	recexprlemell 7563	. . . . . . . . . 10 |- ( r e. ( 1st ` B ) <-> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8	6, 7	sylibr 133	. . . . . . . . 9 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> r e. ( 1st ` B ) )
9	8	adantll 468	. . . . . . . 8 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> r e. ( 1st ` B ) )
10	5, 9	jca 304	. . . . . . 7 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( q <Q r /\ r e. ( 1st ` B ) ) )
11	10	expcom 115	. . . . . 6 |- ( ( *Q ` y ) e. ( 2nd ` A ) -> ( ( q <Q r /\ r <Q y ) -> ( q <Q r /\ r e. ( 1st ` B ) ) ) )
12	11	reximdv 2567	. . . . 5 |- ( ( *Q ` y ) e. ( 2nd ` A ) -> ( E. r e. Q. ( q <Q r /\ r <Q y ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) ) )
13	4, 12	mpan9 279	. . . 4 |- ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
14	13	exlimiv 1586	. . 3 |- ( E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
15	2, 14	sylbi 120	. 2 |- ( q e. ( 1st ` B ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
16	15	3ad2ant3 1010	1 |- ( ( A e. P. /\ q e. Q. /\ q e. ( 1st ` B ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   *Qcrq 7225   <Q cltq 7226   P.cnp 7232

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294

This theorem is referenced by:  recexprlemrnd  7570