Theorem recexprlemopu 7582

Description: The upper cut of B is open. Lemma for recexpr 7593. (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 7578	. . 3 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
3		ltbtwnnqq 7370	. . . . . 6 |- ( y <Q r <-> E. q e. Q. ( y <Q q /\ q <Q r ) )
4	3	biimpi 119	. . . . 5 |- ( y <Q r -> E. q e. Q. ( y <Q q /\ q <Q r ) )
5		simplr 525	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q <Q r )
6		19.8a 1583	. . . . . . . . . 10 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
7	1	recexprlemelu 7578	. . . . . . . . . 10 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
8	6, 7	sylibr 133	. . . . . . . . 9 |- ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
9	8	adantlr 474	. . . . . . . 8 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> q e. ( 2nd ` B ) )
10	5, 9	jca 304	. . . . . . 7 |- ( ( ( y <Q q /\ q <Q r ) /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) )
11	10	expcom 115	. . . . . 6 |- ( ( *Q ` y ) e. ( 1st ` A ) -> ( ( y <Q q /\ q <Q r ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
12	11	reximdv 2571	. . . . 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 279	. . . 4 |- ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
14	13	exlimiv 1591	. . 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 120	. 2 |- ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
16	15	3ad2ant3 1015	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 103   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   <.cop 3584   class class class wbr 3987   ` cfv 5196   1stc1st 6115   2ndc2nd 6116   Q.cnq 7235   *Qcrq 7239   <Q cltq 7240   P.cnp 7246

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-oadd 6397  df-omul 6398  df-er 6511  df-ec 6513  df-qs 6517  df-ni 7259  df-pli 7260  df-mi 7261  df-lti 7262  df-plpq 7299  df-mpq 7300  df-enq 7302  df-nqqs 7303  df-plqqs 7304  df-mqqs 7305  df-1nqqs 7306  df-rq 7307  df-ltnqqs 7308

This theorem is referenced by:  recexprlemrnd  7584