Theorem recexprlemopl 7956

Description: The lower cut of B is open. Lemma for recexpr 7969. (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 7953	. . 3 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
3		ltbtwnnqq 7746	. . . . . 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 1639	. . . . . . . . . 10 |- ( ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> E. y ( r <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
7	1	recexprlemell 7953	. . . . . . . . . 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 2645	. . . . 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 1647	. . 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 1047	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 1005   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   E.wrex 2523   <.cop 3697   class class class wbr 4114   ` cfv 5357   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   *Qcrq 7615   <Q cltq 7616   P.cnp 7622

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  recexprlemrnd  7960