Theorem recexprlemopl 7687

Description: The lower cut of B is open. Lemma for recexpr 7700. (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 7684	. . 3 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
3		ltbtwnnqq 7477	. . . . . 6 |- ( q <Q y <-> E. r e. Q. ( q <Q r /\ r <Q y ) )
4	3	biimpi 120	. . . . 5 |- ( q <Q y -> E. r e. Q. ( q <Q r /\ r <Q y ) )
5		simpll 527	. . . . . . . 8 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> q <Q r )
6		19.8a 1601	. . . . . . . . . 10 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
7	1	recexprlemell 7684	. . . . . . . . . 10 |- ( r e. ( 1st ` B ) <-> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
8	6, 7	sylibr 134	. . . . . . . . 9 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> r e. ( 1st ` B ) )
9	8	adantll 476	. . . . . . . 8 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> r e. ( 1st ` B ) )
10	5, 9	jca 306	. . . . . . 7 |- ( ( ( q <Q r /\ r <Q y ) /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> ( q <Q r /\ r e. ( 1st ` B ) ) )
11	10	expcom 116	. . . . . 6 |- ( ( *Q ` y ) e. ( 2nd ` A ) -> ( ( q <Q r /\ r <Q y ) -> ( q <Q r /\ r e. ( 1st ` B ) ) ) )
12	11	reximdv 2595	. . . . 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 281	. . . 4 |- ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
14	13	exlimiv 1609	. . 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 121	. 2 |- ( q e. ( 1st ` B ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` B ) ) )
16	15	3ad2ant3 1022	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 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   E.wrex 2473   <.cop 3622   class class class wbr 4030   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   *Qcrq 7346   <Q cltq 7347   P.cnp 7353

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415

This theorem is referenced by:  recexprlemrnd  7691