Theorem recexprlemopu 7984

Description: The upper cut of B is open. Lemma for recexpr 7995. (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 7980	. . 3 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
3		ltbtwnnqq 7772	. . . . . 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 533	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q <Q r )
6		19.8a 1643	. . . . . . . . . 10 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
7	1	recexprlemelu 7980	. . . . . . . . . 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 481	. . . . . . . 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 2651	. . . . 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 1651	. . 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 1051	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 1009   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   *Qcrq 7641   <Q cltq 7642   P.cnp 7648

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710

This theorem is referenced by:  recexprlemrnd  7986