Theorem recexprlemopl 7546

Description: The lower cut of B is open. Lemma for recexpr 7559. (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 7543	. . 3 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
3		ltbtwnnqq 7336	. . . . . 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 1570	. . . . . . . . . 10 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
7	1	recexprlemell 7543	. . . . . . . . . 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 2558	. . . . 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 1578	. . 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 1005	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 963   = wceq 1335   E.wex 1472   e. wcel 2128   {cab 2143   E.wrex 2436   <.cop 3563   class class class wbr 3966   ` cfv 5171   1stc1st 6087   2ndc2nd 6088   Q.cnq 7201   *Qcrq 7205   <Q cltq 7206   P.cnp 7212

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-eprel 4250  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-1o 6364  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-pli 7226  df-mi 7227  df-lti 7228  df-plpq 7265  df-mpq 7266  df-enq 7268  df-nqqs 7269  df-plqqs 7270  df-mqqs 7271  df-1nqqs 7272  df-rq 7273  df-ltnqqs 7274

This theorem is referenced by:  recexprlemrnd  7550