Theorem recexprlemell 7542

Description: Membership in the lower cut of B. Lemma for recexpr 7558. (Contributed by Jim Kingdon, 27-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
recexprlemell	|- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem recexprlemell

Step	Hyp	Ref	Expression
1		elex 2723	. 2 |- ( C e. ( 1st ` B ) -> C e. _V )
2		ltrelnq 7285	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4638	. . . . . 6 |- ( C <Q y -> ( C e. Q. /\ y e. Q. ) )
4	3	simpld 111	. . . . 5 |- ( C <Q y -> C e. Q. )
5		elex 2723	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( C <Q y -> C e. _V )
7	6	adantr 274	. . 3 |- ( ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
8	7	exlimiv 1578	. 2 |- ( E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
9		breq1 3968	. . . . 5 |- ( x = C -> ( x <Q y <-> C <Q y ) )
10	9	anbi1d 461	. . . 4 |- ( x = C -> ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
11	10	exbidv 1805	. . 3 |- ( x = C -> ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
12		recexpr.1	. . . . 5 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
13	12	fveq2i 5471	. . . 4 |- ( 1st ` B ) = ( 1st ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. )
14		nqex 7283	. . . . . 6 |- Q. e. _V
15	2	brel 4638	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
16	15	simpld 111	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
17	16	adantr 274	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
18	17	exlimiv 1578	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3203	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4102	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4638	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
22	21	simprd 113	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
23	22	adantr 274	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
24	23	exlimiv 1578	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3203	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4102	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op1st 6094	. . . 4 |- ( 1st ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
28	13, 27	eqtri 2178	. . 3 |- ( 1st ` B ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
29	11, 28	elab2g 2859	. 2 |- ( C e. _V -> ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
30	1, 8, 29	pm5.21nii 694	1 |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   {cab 2143   _Vcvv 2712   <.cop 3563   class class class wbr 3965   ` cfv 5170   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200   *Qcrq 7204   <Q cltq 7205

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-qs 6486  df-ni 7224  df-nqqs 7268  df-ltnqqs 7273

This theorem is referenced by:  recexprlemm  7544  recexprlemopl  7545  recexprlemlol  7546  recexprlemdisj  7550  recexprlemloc  7551  recexprlem1ssl  7553  recexprlemss1l  7555