Theorem recexprlemopu 7459

Description: The upper cut of B is open. Lemma for recexpr 7470. (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 7455	. . 3 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
3		ltbtwnnqq 7247	. . . . . 6 |- ( y <Q r <-> E. q e. Q. ( y <Q q /\ q <Q r ) )
4	3	biimpi 119	. . . . 5 |- ( y <Q r -> E. q e. Q. ( y <Q q /\ q <Q r ) )
5		simplr 520	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q <Q r )
6		19.8a 1570	. . . . . . . . . 10 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
7	1	recexprlemelu 7455	. . . . . . . . . 10 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8	6, 7	sylibr 133	. . . . . . . . 9 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
9	8	adantlr 469	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
10	5, 9	jca 304	. . . . . . 7 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) )
11	10	expcom 115	. . . . . 6 |- ( ( *Q ` y ) e. ( 1st ` A ) -> ( ( y <Q q /\ q <Q r ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
12	11	reximdv 2536	. . . . 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 279	. . . 4 |- ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
14	13	exlimiv 1578	. . 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 120	. 2 |- ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
16	15	3ad2ant3 1005	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 103   /\ w3a 963   = wceq 1332   E.wex 1469   e. wcel 1481   {cab 2126   E.wrex 2418   <.cop 3535   class class class wbr 3937   ` cfv 5131   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   *Qcrq 7116   <Q cltq 7117   P.cnp 7123

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185

This theorem is referenced by:  recexprlemrnd  7461