Theorem suplocexprlemlub 8055

Description: Lemma for suplocexpr 8056. 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 110	. . . 4 |- ( ( ph /\ y <P B ) -> y <P B )
2		ltrelpr 7836	. . . . . . . 8 |- <P C_ ( P. X. P. )
3	2	brel 4807	. . . . . . 7 |- ( y <P B -> ( y e. P. /\ B e. P. ) )
4	3	simpld 112	. . . . . 6 |- ( y <P B -> y e. P. )
5	4	adantl 277	. . . . 5 |- ( ( ph /\ y <P B ) -> y e. P. )
6	3	simprd 114	. . . . . 6 |- ( y <P B -> B e. P. )
7	6	adantl 277	. . . . 5 |- ( ( ph /\ y <P B ) -> B e. P. )
8		ltdfpr 7837	. . . . 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 411	. . . 4 |- ( ( ph /\ y <P B ) -> ( y <P B <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) )
10	1, 9	mpbid 147	. . 3 |- ( ( ph /\ y <P B ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) )
11		simprrr 542	. . . . . 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 5678	. . . . . . . . 9 |- ( 1st ` B ) = ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
14		npex 7804	. . . . . . . . . . . . 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 8046	. . . . . . . . . . . 12 |- ( ph -> A C_ P. )
20	15, 19	ssexd 4255	. . . . . . . . . . 11 |- ( ph -> A e. _V )
21		fo1st 6364	. . . . . . . . . . . . 13 |- 1st : _V -onto-> _V
22		fofun 5596	. . . . . . . . . . . . 13 |- ( 1st : _V -onto-> _V -> Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 |- Fun 1st
24		funimaexg 5445	. . . . . . . . . . . 12 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
25	23, 24	mpan 424	. . . . . . . . . . 11 |- ( A e. _V -> ( 1st " A ) e. _V )
26		uniexg 4565	. . . . . . . . . . 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 7694	. . . . . . . . . . 11 |- Q. e. _V
29	28	rabex 4261	. . . . . . . . . 10 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
30		op1stg 6357	. . . . . . . . . 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 413	. . . . . . . . 9 |- ( ph -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
32	13, 31	eqtrid 2279	. . . . . . . 8 |- ( ph -> ( 1st ` B ) = U. ( 1st " A ) )
33	32	eleq2d 2304	. . . . . . 7 |- ( ph -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
34	33	ad2antrr 488	. . . . . 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 147	. . . . 5 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. U. ( 1st " A ) )
36		suplocexprlemell 8044	. . . . 5 |- ( s e. U. ( 1st " A ) <-> E. z e. A s e. ( 1st ` z ) )
37	35, 36	sylib 122	. . . 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 531	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. Q. )
39	38	ad2antrr 488	. . . . . . . 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 541	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 2nd ` y ) )
41	40	ad2antrr 488	. . . . . . . 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 110	. . . . . . . 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 2593	. . . . . . . 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 1272	. . . . . . 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 495	. . . . . . . 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 494	. . . . . . . . 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 529	. . . . . . . . 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 3243	. . . . . . . 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 7837	. . . . . . . 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 411	. . . . . . 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 167	. . . . . 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 115	. . . . 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 2646	. . . 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 2667	. 2 |- ( ( ph /\ y <P B ) -> E. z e. A y <P z )
56	55	ex 115	1 |- ( ph -> ( y <P B -> E. z e. A y <P z ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   _Vcvv 2815   C_ wss 3214   <.cop 3697   U.cuni 3919   |^|cint 3954   class class class wbr 4114   "cima 4757   Fun wfun 5351   -onto->wfo 5355   ` cfv 5357   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   <Q cltq 7616   P.cnp 7622   <P cltp 7626

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-qs 6786  df-ni 7635  df-nqqs 7679  df-inp 7797  df-iltp 7801

This theorem is referenced by:  suplocexpr  8056