Theorem suplocexprlemub 7785

Description: Lemma for suplocexpr 7787. The putative supremum is an 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
suplocexprlemub	|- ( ph -> A. y e. A -. B <P y )

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

Allowed substitution hints:   B( x, y, z, u)

Proof of Theorem suplocexprlemub

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> B <P y )
2		suplocexpr.m	. . . . . . . 8 |- ( ph -> E. x x e. A )
3		suplocexpr.ub	. . . . . . . 8 |- ( ph -> E. x e. P. A. y e. A y <P x )
4		suplocexpr.loc	. . . . . . . 8 |- ( 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 ) ) )
5		suplocexpr.b	. . . . . . . 8 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
6	2, 3, 4, 5	suplocexprlemex 7784	. . . . . . 7 |- ( ph -> B e. P. )
7	6	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> B e. P. )
8	2, 3, 4	suplocexprlemss 7777	. . . . . . . 8 |- ( ph -> A C_ P. )
9	8	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> A C_ P. )
10		simplr 528	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. A )
11	9, 10	sseldd 3181	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. P. )
12		ltdfpr 7568	. . . . . 6 |- ( ( B e. P. /\ y e. P. ) -> ( B <P y <-> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) )
13	7, 11, 12	syl2anc 411	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> ( B <P y <-> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) )
14	1, 13	mpbid 147	. . . 4 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) )
15		simprrl 539	. . . . . . . 8 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> s e. ( 2nd ` B ) )
16	5	suplocexprlem2b 7776	. . . . . . . . . . 11 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
17	8, 16	syl 14	. . . . . . . . . 10 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
18	17	eleq2d 2263	. . . . . . . . 9 |- ( ph -> ( s e. ( 2nd ` B ) <-> s e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
19	18	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> ( s e. ( 2nd ` B ) <-> s e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
20	15, 19	mpbid 147	. . . . . . 7 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> s e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
21		breq2 4034	. . . . . . . . 9 |- ( u = s -> ( w <Q u <-> w <Q s ) )
22	21	rexbidv 2495	. . . . . . . 8 |- ( u = s -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q s ) )
23	22	elrab 2917	. . . . . . 7 |- ( s e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } <-> ( s e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q s ) )
24	20, 23	sylib 122	. . . . . 6 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> ( s e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q s ) )
25	24	simprd 114	. . . . 5 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> E. w e. |^| ( 2nd " A ) w <Q s )
26		simprrr 540	. . . . . . . 8 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> s e. ( 1st ` y ) )
27	26	adantr 276	. . . . . . 7 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> s e. ( 1st ` y ) )
28		simprr 531	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> w <Q s )
29	11	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> y e. P. )
30		prop 7537	. . . . . . . . . 10 |- ( y e. P. -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
32		eleq2 2257	. . . . . . . . . 10 |- ( t = ( 2nd ` y ) -> ( w e. t <-> w e. ( 2nd ` y ) ) )
33		simprl 529	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> w e. |^| ( 2nd " A ) )
34		vex 2763	. . . . . . . . . . . 12 |- w e. _V
35	34	elint2 3878	. . . . . . . . . . 11 |- ( w e. |^| ( 2nd " A ) <-> A. t e. ( 2nd " A ) w e. t )
36	33, 35	sylib 122	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> A. t e. ( 2nd " A ) w e. t )
37		fo2nd 6213	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
38		fofun 5478	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
40		vex 2763	. . . . . . . . . . . . 13 |- y e. _V
41		fof 5477	. . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 |- 2nd : _V --> _V
43	42	fdmi 5412	. . . . . . . . . . . . 13 |- dom 2nd = _V
44	40, 43	eleqtrri 2269	. . . . . . . . . . . 12 |- y e. dom 2nd
45		funfvima 5791	. . . . . . . . . . . 12 |- ( ( Fun 2nd /\ y e. dom 2nd ) -> ( y e. A -> ( 2nd ` y ) e. ( 2nd " A ) ) )
46	39, 44, 45	mp2an 426	. . . . . . . . . . 11 |- ( y e. A -> ( 2nd ` y ) e. ( 2nd " A ) )
47	46	ad4antlr 495	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> ( 2nd ` y ) e. ( 2nd " A ) )
48	32, 36, 47	rspcdva 2870	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> w e. ( 2nd ` y ) )
49		prcunqu 7547	. . . . . . . . 9 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ w e. ( 2nd ` y ) ) -> ( w <Q s -> s e. ( 2nd ` y ) ) )
50	31, 48, 49	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> ( w <Q s -> s e. ( 2nd ` y ) ) )
51	28, 50	mpd 13	. . . . . . 7 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> s e. ( 2nd ` y ) )
52	27, 51	jca 306	. . . . . 6 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> ( s e. ( 1st ` y ) /\ s e. ( 2nd ` y ) ) )
53		simplrl 535	. . . . . . 7 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> s e. Q. )
54		prdisj 7554	. . . . . . 7 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ s e. Q. ) -> -. ( s e. ( 1st ` y ) /\ s e. ( 2nd ` y ) ) )
55	31, 53, 54	syl2anc 411	. . . . . 6 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> -. ( s e. ( 1st ` y ) /\ s e. ( 2nd ` y ) ) )
56	52, 55	pm2.21fal 1384	. . . . 5 |- ( ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q s ) ) -> F. )
57	25, 56	rexlimddv 2616	. . . 4 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> F. )
58	14, 57	rexlimddv 2616	. . 3 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> F. )
59	58	inegd 1383	. 2 |- ( ( ph /\ y e. A ) -> -. B <P y )
60	59	ralrimiva 2567	1 |- ( ph -> A. y e. A -. B <P y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   F. wfal 1369   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3154   <.cop 3622   U.cuni 3836   |^|cint 3871   class class class wbr 4030   dom cdm 4660   "cima 4663   Fun wfun 5249   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   <Q cltq 7347   P.cnp 7353   <P cltp 7357

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iltp 7532

This theorem is referenced by:  suplocexpr  7787