Theorem recexprlemopu 6725

Description: The upper cut of B is open. Lemma for recexpr 6736. (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
recexprlemopu	|- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )

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

Proof of Theorem recexprlemopu

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	recexprlemelu 6721	. . 3 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
3		ltbtwnnqq 6513	. . . . . 6 |- ( y <Q r <-> E. q e. Q. ( y <Q q /\ q <Q r ) )
4	3	biimpi 113	. . . . 5 |- ( y <Q r -> E. q e. Q. ( y <Q q /\ q <Q r ) )
5		simplr 482	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q <Q r )
6		19.8a 1482	. . . . . . . . . 10 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
7	1	recexprlemelu 6721	. . . . . . . . . 10 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8	6, 7	sylibr 137	. . . . . . . . 9 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
9	8	adantlr 446	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
10	5, 9	jca 290	. . . . . . 7 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) )
11	10	expcom 109	. . . . . 6 |- ( ( *Q ` y ) e. ( 1st ` A ) -> ( ( y <Q q /\ q <Q r ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
12	11	reximdv 2420	. . . . 5 |- ( ( *Q ` y ) e. ( 1st ` A ) -> ( E. q e. Q. ( y <Q q /\ q <Q r ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
13	4, 12	mpan9 265	. . . 4 |- ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
14	13	exlimiv 1489	. . 3 |- ( E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
15	2, 14	sylbi 114	. 2 |- ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
16	15	3ad2ant3 927	1 |- ( ( A e. P. /\ r e. Q. /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   <.cop 3378   class class class wbr 3764   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   *Qcrq 6382   <Q cltq 6383   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451

This theorem is referenced by:  recexprlemrnd  6727