Theorem suplocexprlemlub 7532

Description: Lemma for suplocexpr 7533. 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 7313	. . . . . . . 8 |- <P C_ ( P. X. P. )
3	2	brel 4591	. . . . . . 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 7314	. . . . 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 408	. . . 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 529	. . . . . 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 5424	. . . . . . . . 9 |- ( 1st ` B ) = ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
14		npex 7281	. . . . . . . . . . . . 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 7523	. . . . . . . . . . . 12 |- ( ph -> A C_ P. )
20	15, 19	ssexd 4068	. . . . . . . . . . 11 |- ( ph -> A e. _V )
21		fo1st 6055	. . . . . . . . . . . . 13 |- 1st : _V -onto-> _V
22		fofun 5346	. . . . . . . . . . . . 13 |- ( 1st : _V -onto-> _V -> Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 |- Fun 1st
24		funimaexg 5207	. . . . . . . . . . . 12 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
25	23, 24	mpan 420	. . . . . . . . . . 11 |- ( A e. _V -> ( 1st " A ) e. _V )
26		uniexg 4361	. . . . . . . . . . 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 7171	. . . . . . . . . . 11 |- Q. e. _V
29	28	rabex 4072	. . . . . . . . . 10 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
30		op1stg 6048	. . . . . . . . . 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 409	. . . . . . . . 9 |- ( ph -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
32	13, 31	syl5eq 2184	. . . . . . . 8 |- ( ph -> ( 1st ` B ) = U. ( 1st " A ) )
33	32	eleq2d 2209	. . . . . . 7 |- ( ph -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
34	33	ad2antrr 479	. . . . . 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 7521	. . . . 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 520	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. Q. )
39	38	ad2antrr 479	. . . . . . . 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 528	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 2nd ` y ) )
41	40	ad2antrr 479	. . . . . . . 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 2481	. . . . . . . 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 1214	. . . . . . 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 486	. . . . . . . 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 485	. . . . . . . . 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 519	. . . . . . . . 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 3098	. . . . . . . 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 7314	. . . . . . . 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 408	. . . . . . 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 2534	. . . 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 2554	. 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 697   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   _Vcvv 2686   C_ wss 3071   <.cop 3530   U.cuni 3736   |^|cint 3771   class class class wbr 3929   "cima 4542   Fun wfun 5117   -onto->wfo 5121   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   <Q cltq 7093   P.cnp 7099   <P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-qs 6435  df-ni 7112  df-nqqs 7156  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexpr  7533