Theorem ltexprlemell 7518

Description: Element in lower cut of the constructed difference. Lemma for ltexpri 7533. (Contributed by Jim Kingdon, 21-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemell	|- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )

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

Proof of Theorem ltexprlemell

Step	Hyp	Ref	Expression
1		oveq2 5832	. . . . 5 |- ( x = q -> ( y +Q x ) = ( y +Q q ) )
2	1	eleq1d 2226	. . . 4 |- ( x = q -> ( ( y +Q x ) e. ( 1st ` B ) <-> ( y +Q q ) e. ( 1st ` B ) ) )
3	2	anbi2d 460	. . 3 |- ( x = q -> ( ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) <-> ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
4	3	exbidv 1805	. 2 |- ( x = q -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
5		ltexprlem.1	. . . 4 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
6	5	fveq2i 5471	. . 3 |- ( 1st ` C ) = ( 1st ` <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. )
7		nqex 7283	. . . . 5 |- Q. e. _V
8	7	rabex 4108	. . . 4 |- { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } e. _V
9	7	rabex 4108	. . . 4 |- { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } e. _V
10	8, 9	op1st 6094	. . 3 |- ( 1st ` <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >. ) = { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) }
11	6, 10	eqtri 2178	. 2 |- ( 1st ` C ) = { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) }
12	4, 11	elrab2 2871	1 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   {crab 2439   <.cop 3563   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200   +Q cplq 7202

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-ov 5827  df-1st 6088  df-qs 6486  df-ni 7224  df-nqqs 7268

This theorem is referenced by:  ltexprlemm  7520  ltexprlemopl  7521  ltexprlemlol  7522  ltexprlemdisj  7526  ltexprlemloc  7527  ltexprlemfl  7529  ltexprlemrl  7530