Theorem recexprlemupu 7943

Description: The upper cut of B is upper. Lemma for recexpr 7953. (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
recexprlemupu	|- ( ( A e. P. /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemupu

Step	Hyp	Ref	Expression
1		ltsonq 7713	. . . . . . . . 9 |- <Q Or Q.
2		ltrelnq 7680	. . . . . . . . 9 |- <Q C_ ( Q. X. Q. )
3	1, 2	sotri 5158	. . . . . . . 8 |- ( ( y <Q q /\ q <Q r ) -> y <Q r )
4	3	expcom 116	. . . . . . 7 |- ( q <Q r -> ( y <Q q -> y <Q r ) )
5	4	anim1d 336	. . . . . 6 |- ( q <Q r -> ( ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
6	5	eximdv 1929	. . . . 5 |- ( q <Q r -> ( E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
7		recexpr.1	. . . . . 6 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
8	7	recexprlemelu 7938	. . . . 5 |- ( q e. ( 2nd ` B ) <-> E. y ( y <Q q /\ ( *Q ` y ) e. ( 1st ` A ) ) )
9	7	recexprlemelu 7938	. . . . 5 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
10	6, 8, 9	3imtr4g 205	. . . 4 |- ( q <Q r -> ( q e. ( 2nd ` B ) -> r e. ( 2nd ` B ) ) )
11	10	imp 124	. . 3 |- ( ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) )
12	11	rexlimivw 2656	. 2 |- ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) )
13	12	a1i 9	1 |- ( ( A e. P. /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   E.wrex 2521   <.cop 3692   class class class wbr 4109   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   *Qcrq 7599   <Q cltq 7600   P.cnp 7606

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-mi 7621  df-lti 7622  df-enq 7662  df-nqqs 7663  df-ltnqqs 7668

This theorem is referenced by:  recexprlemrnd  7944