Theorem suplocexprlemub 7713

Description: Lemma for suplocexpr 7715. 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 7712	. . . . . . 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 7705	. . . . . . . 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 3156	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. P. )
12		ltdfpr 7496	. . . . . 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 7704	. . . . . . . . . . 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 2247	. . . . . . . . 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 4004	. . . . . . . . 9 |- ( u = s -> ( w <Q u <-> w <Q s ) )
22	21	rexbidv 2478	. . . . . . . 8 |- ( u = s -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q s ) )
23	22	elrab 2893	. . . . . . 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 7465	. . . . . . . . . 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 2241	. . . . . . . . . 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 2740	. . . . . . . . . . . 12 |- w e. _V
35	34	elint2 3849	. . . . . . . . . . 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 6153	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
38		fofun 5435	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
40		vex 2740	. . . . . . . . . . . . 13 |- y e. _V
41		fof 5434	. . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 |- 2nd : _V --> _V
43	42	fdmi 5369	. . . . . . . . . . . . 13 |- dom 2nd = _V
44	40, 43	eleqtrri 2253	. . . . . . . . . . . 12 |- y e. dom 2nd
45		funfvima 5743	. . . . . . . . . . . 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 2846	. . . . . . . . 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 7475	. . . . . . . . 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 7482	. . . . . . 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 1373	. . . . 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 2599	. . . 4 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> F. )
58	14, 57	rexlimddv 2599	. . 3 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> F. )
59	58	inegd 1372	. 2 |- ( ( ph /\ y e. A ) -> -. B <P y )
60	59	ralrimiva 2550	1 |- ( ph -> A. y e. A -. B <P y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   F. wfal 1358   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2737   C_ wss 3129   <.cop 3594   U.cuni 3807   |^|cint 3842   class class class wbr 4000   dom cdm 4623   "cima 4626   Fun wfun 5206   -->wf 5208   -onto->wfo 5210   ` cfv 5212   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   <Q cltq 7275   P.cnp 7281   <P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460

This theorem is referenced by:  suplocexpr  7715