Theorem recexprlemell 7735

Description: Membership in the lower cut of B. Lemma for recexpr 7751. (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 2783	. 2 |- ( C e. ( 1st ` B ) -> C e. _V )
2		ltrelnq 7478	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
3	2	brel 4727	. . . . . 6 |- ( C <Q y -> ( C e. Q. /\ y e. Q. ) )
4	3	simpld 112	. . . . 5 |- ( C <Q y -> C e. Q. )
5		elex 2783	. . . . 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 1621	. 2 |- ( E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> C e. _V )
9		breq1 4047	. . . . 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 1848	. . 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 5579	. . . 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 7476	. . . . . 6 |- Q. e. _V
15	2	brel 4727	. . . . . . . . . 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 1621	. . . . . . 7 |- ( E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) -> x e. Q. )
19	18	abssi 3268	. . . . . 6 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } C_ Q.
20	14, 19	ssexi 4182	. . . . 5 |- { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } e. _V
21	2	brel 4727	. . . . . . . . . 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 1621	. . . . . . 7 |- ( E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) -> x e. Q. )
25	24	abssi 3268	. . . . . 6 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } C_ Q.
26	14, 25	ssexi 4182	. . . . 5 |- { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } e. _V
27	20, 26	op1st 6232	. . . 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 2226	. . 3 |- ( 1st ` B ) = { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) }
29	11, 28	elab2g 2920	. 2 |- ( C e. _V -> ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
30	1, 8, 29	pm5.21nii 706	1 |- ( C e. ( 1st ` B ) <-> E. y ( C <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   _Vcvv 2772   <.cop 3636   class class class wbr 4044   ` cfv 5271   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   *Qcrq 7397   <Q cltq 7398

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-qs 6626  df-ni 7417  df-nqqs 7461  df-ltnqqs 7466

This theorem is referenced by:  recexprlemm  7737  recexprlemopl  7738  recexprlemlol  7739  recexprlemdisj  7743  recexprlemloc  7744  recexprlem1ssl  7746  recexprlemss1l  7748