Theorem suplocexprlemub 7531

Description: Lemma for suplocexpr 7533. 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 109	. . . . 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 7530	. . . . . . 7 |- ( ph -> B e. P. )
7	6	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> B e. P. )
8	2, 3, 4	suplocexprlemss 7523	. . . . . . . 8 |- ( ph -> A C_ P. )
9	8	ad2antrr 479	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> A C_ P. )
10		simplr 519	. . . . . . 7 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. A )
11	9, 10	sseldd 3098	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> y e. P. )
12		ltdfpr 7314	. . . . . 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 408	. . . . 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 146	. . . 4 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) )
15		simprrl 528	. . . . . . . 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 7522	. . . . . . . . . . 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 2209	. . . . . . . . 9 |- ( ph -> ( s e. ( 2nd ` B ) <-> s e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
19	18	ad3antrrr 483	. . . . . . . 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 146	. . . . . . 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 3933	. . . . . . . . 9 |- ( u = s -> ( w <Q u <-> w <Q s ) )
22	21	rexbidv 2438	. . . . . . . 8 |- ( u = s -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q s ) )
23	22	elrab 2840	. . . . . . 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 121	. . . . . 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 113	. . . . 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 529	. . . . . . . 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 274	. . . . . . 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 521	. . . . . . . 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 479	. . . . . . . . . 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 7283	. . . . . . . . . 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 2203	. . . . . . . . . 10 |- ( t = ( 2nd ` y ) -> ( w e. t <-> w e. ( 2nd ` y ) ) )
33		simprl 520	. . . . . . . . . . 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 2689	. . . . . . . . . . . 12 |- w e. _V
35	34	elint2 3778	. . . . . . . . . . 11 |- ( w e. |^| ( 2nd " A ) <-> A. t e. ( 2nd " A ) w e. t )
36	33, 35	sylib 121	. . . . . . . . . 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 6056	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
38		fofun 5346	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
40		vex 2689	. . . . . . . . . . . . 13 |- y e. _V
41		fof 5345	. . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 |- 2nd : _V --> _V
43	42	fdmi 5280	. . . . . . . . . . . . 13 |- dom 2nd = _V
44	40, 43	eleqtrri 2215	. . . . . . . . . . . 12 |- y e. dom 2nd
45		funfvima 5649	. . . . . . . . . . . 12 |- ( ( Fun 2nd /\ y e. dom 2nd ) -> ( y e. A -> ( 2nd ` y ) e. ( 2nd " A ) ) )
46	39, 44, 45	mp2an 422	. . . . . . . . . . 11 |- ( y e. A -> ( 2nd ` y ) e. ( 2nd " A ) )
47	46	ad4antlr 486	. . . . . . . . . 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 2794	. . . . . . . . 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 7293	. . . . . . . . 9 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ w e. ( 2nd ` y ) ) -> ( w <Q s -> s e. ( 2nd ` y ) ) )
50	31, 48, 49	syl2anc 408	. . . . . . . 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 304	. . . . . 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 524	. . . . . . 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 7300	. . . . . . 7 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ s e. Q. ) -> -. ( s e. ( 1st ` y ) /\ s e. ( 2nd ` y ) ) )
55	31, 53, 54	syl2anc 408	. . . . . 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 1351	. . . . 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 2554	. . . 4 |- ( ( ( ( ph /\ y e. A ) /\ B <P y ) /\ ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` y ) ) ) ) -> F. )
58	14, 57	rexlimddv 2554	. . 3 |- ( ( ( ph /\ y e. A ) /\ B <P y ) -> F. )
59	58	inegd 1350	. 2 |- ( ( ph /\ y e. A ) -> -. B <P y )
60	59	ralrimiva 2505	1 |- ( ph -> A. y e. A -. B <P y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   F. wfal 1336   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   dom cdm 4539   "cima 4542   Fun wfun 5117   -->wf 5119   -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-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  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-ne 2309  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-nul 3364  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-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  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-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexpr  7533