Theorem recexprlemopu 7814

Description: The upper cut of B is open. Lemma for recexpr 7825. (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 7810	. . 3 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
3		ltbtwnnqq 7602	. . . . . 6 |- ( y <Q r <-> E. q e. Q. ( y <Q q /\ q <Q r ) )
4	3	biimpi 120	. . . . 5 |- ( y <Q r -> E. q e. Q. ( y <Q q /\ q <Q r ) )
5		simplr 528	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q <Q r )
6		19.8a 1636	. . . . . . . . . 10 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
7	1	recexprlemelu 7810	. . . . . . . . . 10 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8	6, 7	sylibr 134	. . . . . . . . 9 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
9	8	adantlr 477	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
10	5, 9	jca 306	. . . . . . 7 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) )
11	10	expcom 116	. . . . . 6 |- ( ( *Q ` y ) e. ( 1st ` A ) -> ( ( y <Q q /\ q <Q r ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
12	11	reximdv 2631	. . . . 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 281	. . . 4 |- ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
14	13	exlimiv 1644	. . 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 121	. 2 |- ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
16	15	3ad2ant3 1044	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 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   <.cop 3669   class class class wbr 4083   ` cfv 5318   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   *Qcrq 7471   <Q cltq 7472   P.cnp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540

This theorem is referenced by:  recexprlemrnd  7816