Theorem recexprlemopl 7738

Description: The lower cut of B is open. Lemma for recexpr 7751. (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 7735	. . 3 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
3		ltbtwnnqq 7528	. . . . . 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 1613	. . . . . . . . . 10 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
7	1	recexprlemell 7735	. . . . . . . . . 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 2607	. . . . 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 1621	. . 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 1023	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 981   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   <.cop 3636   class class class wbr 4044   ` cfv 5271   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   *Qcrq 7397   <Q cltq 7398   P.cnp 7404

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466

This theorem is referenced by:  recexprlemrnd  7742