Theorem suplocexprlemlub 7556

Description: Lemma for suplocexpr 7557. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	|- ( ph -> E. x x e. A )
suplocexpr.ub	|- ( ph -> E. x e. P. A. y e. A y <P x )
suplocexpr.loc	|- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
suplocexpr.b	|- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.

Assertion

Ref	Expression
suplocexprlemlub	|- ( ph -> ( y <P B -> E. z e. A y <P z ) )

Distinct variable groups:   y, A, z   x, A, y   z, B   ph, y, z   ph, x

Allowed substitution hints:   ph( w, u)   A( w, u)   B( x, y, w, u)

Proof of Theorem suplocexprlemlub

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 |- ( ( ph /\ y <P B ) -> y <P B )
2		ltrelpr 7337	. . . . . . . 8 |- <P C_ ( P. X. P. )
3	2	brel 4599	. . . . . . 7 |- ( y <P B -> ( y e. P. /\ B e. P. ) )
4	3	simpld 111	. . . . . 6 |- ( y <P B -> y e. P. )
5	4	adantl 275	. . . . 5 |- ( ( ph /\ y <P B ) -> y e. P. )
6	3	simprd 113	. . . . . 6 |- ( y <P B -> B e. P. )
7	6	adantl 275	. . . . 5 |- ( ( ph /\ y <P B ) -> B e. P. )
8		ltdfpr 7338	. . . . 5 |- ( ( y e. P. /\ B e. P. ) -> ( y <P B <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) )
9	5, 7, 8	syl2anc 409	. . . 4 |- ( ( ph /\ y <P B ) -> ( y <P B <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) )
10	1, 9	mpbid 146	. . 3 |- ( ( ph /\ y <P B ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) )
11		simprrr 530	. . . . . 6 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 1st ` B ) )
12		suplocexpr.b	. . . . . . . . . 10 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
13	12	fveq2i 5432	. . . . . . . . 9 |- ( 1st ` B ) = ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
14		npex 7305	. . . . . . . . . . . . 13 |- P. e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> P. e. _V )
16		suplocexpr.m	. . . . . . . . . . . . 13 |- ( ph -> E. x x e. A )
17		suplocexpr.ub	. . . . . . . . . . . . 13 |- ( ph -> E. x e. P. A. y e. A y <P x )
18		suplocexpr.loc	. . . . . . . . . . . . 13 |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
19	16, 17, 18	suplocexprlemss 7547	. . . . . . . . . . . 12 |- ( ph -> A C_ P. )
20	15, 19	ssexd 4076	. . . . . . . . . . 11 |- ( ph -> A e. _V )
21		fo1st 6063	. . . . . . . . . . . . 13 |- 1st : _V -onto-> _V
22		fofun 5354	. . . . . . . . . . . . 13 |- ( 1st : _V -onto-> _V -> Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 |- Fun 1st
24		funimaexg 5215	. . . . . . . . . . . 12 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
25	23, 24	mpan 421	. . . . . . . . . . 11 |- ( A e. _V -> ( 1st " A ) e. _V )
26		uniexg 4369	. . . . . . . . . . 11 |- ( ( 1st " A ) e. _V -> U. ( 1st " A ) e. _V )
27	20, 25, 26	3syl 17	. . . . . . . . . 10 |- ( ph -> U. ( 1st " A ) e. _V )
28		nqex 7195	. . . . . . . . . . 11 |- Q. e. _V
29	28	rabex 4080	. . . . . . . . . 10 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
30		op1stg 6056	. . . . . . . . . 10 |- ( ( U. ( 1st " A ) e. _V /\ { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V ) -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
31	27, 29, 30	sylancl 410	. . . . . . . . 9 |- ( ph -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
32	13, 31	syl5eq 2185	. . . . . . . 8 |- ( ph -> ( 1st ` B ) = U. ( 1st " A ) )
33	32	eleq2d 2210	. . . . . . 7 |- ( ph -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
34	33	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
35	11, 34	mpbid 146	. . . . 5 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. U. ( 1st " A ) )
36		suplocexprlemell 7545	. . . . 5 |- ( s e. U. ( 1st " A ) <-> E. z e. A s e. ( 1st ` z ) )
37	35, 36	sylib 121	. . . 4 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> E. z e. A s e. ( 1st ` z ) )
38		simprl 521	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. Q. )
39	38	ad2antrr 480	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. Q. )
40		simprrl 529	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 2nd ` y ) )
41	40	ad2antrr 480	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. ( 2nd ` y ) )
42		simpr 109	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. ( 1st ` z ) )
43		rspe 2484	. . . . . . . 8 |- ( ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) )
44	39, 41, 42, 43	syl12anc 1215	. . . . . . 7 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) )
45	4	ad4antlr 487	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> y e. P. )
46	19	ad4antr 486	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> A C_ P. )
47		simplr 520	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> z e. A )
48	46, 47	sseldd 3103	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> z e. P. )
49		ltdfpr 7338	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( y <P z <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) )
50	45, 48, 49	syl2anc 409	. . . . . . 7 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> ( y <P z <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) )
51	44, 50	mpbird 166	. . . . . 6 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> y <P z )
52	51	ex 114	. . . . 5 |- ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) -> ( s e. ( 1st ` z ) -> y <P z ) )
53	52	reximdva 2537	. . . 4 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> ( E. z e. A s e. ( 1st ` z ) -> E. z e. A y <P z ) )
54	37, 53	mpd 13	. . 3 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> E. z e. A y <P z )
55	10, 54	rexlimddv 2557	. 2 |- ( ( ph /\ y <P B ) -> E. z e. A y <P z )
56	55	ex 114	1 |- ( ph -> ( y <P B -> E. z e. A y <P z ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   _Vcvv 2689   C_ wss 3076   <.cop 3535   U.cuni 3744   |^|cint 3779   class class class wbr 3937   "cima 4550   Fun wfun 5125   -onto->wfo 5129   ` cfv 5131   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   <Q cltq 7117   P.cnp 7123   <P cltp 7127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-qs 6443  df-ni 7136  df-nqqs 7180  df-inp 7298  df-iltp 7302

This theorem is referenced by:  suplocexpr  7557