Theorem recexprlemell 7937

Description: Membership in the lower cut of B. Lemma for recexpr 7953. (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 2825	. 2 |- ( C e. ( 1st ` B ) -> C e. _V )
2		ltrelnq 7680	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4802	. . . . . 6 |- ( C <Q y -> ( C e. Q. /\ y e. Q. ) )
4	3	simpld 112	. . . . 5 |- ( C <Q y -> C e. Q. )
5		elex 2825	. . . . 5 |- ( C e. Q. -> C e. _V )
6	4, 5	syl 14	. . . 4 |- ( C <Q y -> C e. _V )
7	6	adantr 276	. . 3 |- ( ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
8	7	exlimiv 1647	. 2 |- ( E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
9		breq1 4112	. . . . 5 |- ( x = C -> ( x <Q y <-> C <Q y ) )
10	9	anbi1d 465	. . . 4 |- ( x = C -> ( ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
11	10	exbidv 1874	. . 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 5673	. . . 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 7678	. . . . . 6 |- Q. e. _V
15	2	brel 4802	. . . . . . . . . 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 1647	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3313	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4248	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4802	. . . . . . . . . 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 1647	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3313	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4248	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op1st 6340	. . . 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 2253	. . 3 |- ( 1st ` B ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
29	11, 28	elab2g 2964	. 2 |- ( C e. _V -> ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
30	1, 8, 29	pm5.21nii 712	1 |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   _Vcvv 2813   <.cop 3692   class class class wbr 4109   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   *Qcrq 7599   <Q cltq 7600

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-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-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-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-id 4414  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-1st 6334  df-qs 6773  df-ni 7619  df-nqqs 7663  df-ltnqqs 7668

This theorem is referenced by:  recexprlemm  7939  recexprlemopl  7940  recexprlemlol  7941  recexprlemdisj  7945  recexprlemloc  7946  recexprlem1ssl  7948  recexprlemss1l  7950