Theorem recexprlemelu 7685

Description: Membership in the upper cut of B. Lemma for recexpr 7700. (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
recexprlemelu	|- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )

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

Proof of Theorem recexprlemelu

Step	Hyp	Ref	Expression
1		elex 2771	. 2 |- ( C e. ( 2nd ` B ) -> C e. _V )
2		ltrelnq 7427	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4712	. . . . . 6 |- ( y <Q C -> ( y e. Q. /\ C e. Q. ) )
4	3	simprd 114	. . . . 5 |- ( y <Q C -> C e. Q. )
5		elex 2771	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( y <Q C -> C e. _V )
7	6	adantr 276	. . 3 |- ( ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
8	7	exlimiv 1609	. 2 |- ( E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) -> C e. _V )
9		breq2 4034	. . . . 5 |- ( x = C -> ( y <Q x <-> y <Q C ) )
10	9	anbi1d 465	. . . 4 |- ( x = C -> ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
11	10	exbidv 1836	. . 3 |- ( x = C -> ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` 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 5558	. . . 4 |- ( 2nd ` B ) = ( 2nd ` <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >. )
14		nqex 7425	. . . . . 6 |- Q. e. _V
15	2	brel 4712	. . . . . . . . . 10 |- ( x <Q y -> ( x e. Q. /\ y e. Q. ) )
16	15	simpld 112	. . . . . . . . 9 |- ( x <Q y -> x e. Q. )
17	16	adantr 276	. . . . . . . 8 |- ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
18	17	exlimiv 1609	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3255	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4168	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4712	. . . . . . . . . 10 |- ( y <Q x -> ( y e. Q. /\ x e. Q. ) )
22	21	simprd 114	. . . . . . . . 9 |- ( y <Q x -> x e. Q. )
23	22	adantr 276	. . . . . . . 8 |- ( ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
24	23	exlimiv 1609	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3255	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4168	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op2nd 6202	. . . 4 |- ( 2nd ` <. { 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 ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) }
28	13, 27	eqtri 2214	. . 3 |- ( 2nd ` B ) = { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) }
29	11, 28	elab2g 2908	. 2 |- ( C e. _V -> ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
30	1, 8, 29	pm5.21nii 705	1 |- ( C e. ( 2nd ` B ) <-> E. y ( y <Q C /\ ( *Q ` y ) e. ( 1st ` A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   _Vcvv 2760   <.cop 3622   class class class wbr 4030   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   *Qcrq 7346   <Q cltq 7347

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-2nd 6196  df-qs 6595  df-ni 7366  df-nqqs 7410  df-ltnqqs 7415

This theorem is referenced by:  recexprlemm  7686  recexprlemopu  7689  recexprlemupu  7690  recexprlemdisj  7692  recexprlemloc  7693  recexprlem1ssu  7696  recexprlemss1u  7698