Theorem suplocexprlemub 8038

Description: Lemma for suplocexpr 8040. 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 8037	. . . . . . 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 8030	. . . . . . . 8 |- ( ph -> A C_ P. )
9	8	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> A C_ P. )
10		simplr 529	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. A )
11	9, 10	sseldd 3239	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. P. )
12		ltdfpr 7821	. . . . . 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 541	. . . . . . . 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 8029	. . . . . . . . . . 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 2302	. . . . . . . . 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 4113	. . . . . . . . 9 |- ( u = s -> ( w <Q u <-> w <Q s ) )
22	21	rexbidv 2543	. . . . . . . 8 |- ( u = s -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q s ) )
23	22	elrab 2973	. . . . . . 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 542	. . . . . . . 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 533	. . . . . . . 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 7790	. . . . . . . . . 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 2296	. . . . . . . . . 10 |- ( t = ( 2nd ` y ) -> ( w e. t <-> w e. ( 2nd ` y ) ) )
33		simprl 531	. . . . . . . . . . 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 2816	. . . . . . . . . . . 12 |- w e. _V
35	34	elint2 3956	. . . . . . . . . . 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 6352	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
38		fofun 5591	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
40		vex 2816	. . . . . . . . . . . . 13 |- y e. _V
41		fof 5590	. . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 |- 2nd : _V --> _V
43	42	fdmi 5516	. . . . . . . . . . . . 13 |- dom 2nd = _V
44	40, 43	eleqtrri 2308	. . . . . . . . . . . 12 |- y e. dom 2nd
45		funfvima 5918	. . . . . . . . . . . 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 2926	. . . . . . . . 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 7800	. . . . . . . . 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 537	. . . . . . 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 7807	. . . . . . 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 1418	. . . . 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 2665	. . . 4 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> F. )
58	14, 57	rexlimddv 2665	. . 3 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> F. )
59	58	inegd 1417	. 2 |- ( ( ph /\ y e. A ) -> -. B <P y )
60	59	ralrimiva 2615	1 |- ( ph -> A. y e. A -. B <P y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   F. wfal 1403   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   _Vcvv 2813   C_ wss 3211   <.cop 3692   U.cuni 3914   |^|cint 3949   class class class wbr 4109   dom cdm 4749   "cima 4752   Fun wfun 5346   -->wf 5348   -onto->wfo 5350   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   <Q cltq 7600   P.cnp 7606   <P cltp 7610

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  suplocexpr  8040