Theorem suplocexprlemdisj 7832

Description: Lemma for suplocexpr 7837. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-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
suplocexprlemdisj	|- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )

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

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

Proof of Theorem suplocexprlemdisj

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

Step	Hyp	Ref	Expression
1		simprl 529	. . . . 5 |- ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) -> q e. U. ( 1st " A ) )
2		suplocexprlemell 7825	. . . . 5 |- ( q e. U. ( 1st " A ) <-> E. s e. A q e. ( 1st ` s ) )
3	1, 2	sylib 122	. . . 4 |- ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) -> E. s e. A q e. ( 1st ` s ) )
4		simprr 531	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. ( 1st ` s ) )
5		simplrr 536	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. ( 2nd ` B ) )
6		suplocexpr.m	. . . . . . . . . . . . 13 |- ( ph -> E. x x e. A )
7		suplocexpr.ub	. . . . . . . . . . . . 13 |- ( ph -> E. x e. P. A. y e. A y <P x )
8		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 ) ) )
9	6, 7, 8	suplocexprlemss 7827	. . . . . . . . . . . 12 |- ( ph -> A C_ P. )
10	9	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> A C_ P. )
11		suplocexpr.b	. . . . . . . . . . . . 13 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
12	11	suplocexprlem2b 7826	. . . . . . . . . . . 12 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
13	12	eleq2d 2274	. . . . . . . . . . 11 |- ( A C_ P. -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
14	10, 13	syl 14	. . . . . . . . . 10 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
15		breq2 4047	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
16	15	rexbidv 2506	. . . . . . . . . . 11 |- ( u = q -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q q ) )
17	16	elrab 2928	. . . . . . . . . 10 |- ( q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } <-> ( q e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q q ) )
18	14, 17	bitrdi 196	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> ( q e. ( 2nd ` B ) <-> ( q e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q q ) ) )
19	5, 18	mpbid 147	. . . . . . . 8 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> ( q e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q q ) )
20	19	simprd 114	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> E. w e. |^| ( 2nd " A ) w <Q q )
21		simprr 531	. . . . . . . 8 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> w <Q q )
22	10	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> A C_ P. )
23		simplrl 535	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> s e. A )
24	22, 23	sseldd 3193	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> s e. P. )
25		prop 7587	. . . . . . . . . 10 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
26	24, 25	syl 14	. . . . . . . . 9 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
27		eleq2 2268	. . . . . . . . . 10 |- ( t = ( 2nd ` s ) -> ( w e. t <-> w e. ( 2nd ` s ) ) )
28		simprl 529	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> w e. |^| ( 2nd " A ) )
29		vex 2774	. . . . . . . . . . . 12 |- w e. _V
30	29	elint2 3891	. . . . . . . . . . 11 |- ( w e. |^| ( 2nd " A ) <-> A. t e. ( 2nd " A ) w e. t )
31	28, 30	sylib 122	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> A. t e. ( 2nd " A ) w e. t )
32		fo2nd 6243	. . . . . . . . . . . . . 14 |- 2nd : _V -onto-> _V
33		fofun 5498	. . . . . . . . . . . . . 14 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 |- Fun 2nd
35		vex 2774	. . . . . . . . . . . . . 14 |- s e. _V
36		fof 5497	. . . . . . . . . . . . . . . 16 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 |- 2nd : _V --> _V
38	37	fdmi 5432	. . . . . . . . . . . . . 14 |- dom 2nd = _V
39	35, 38	eleqtrri 2280	. . . . . . . . . . . . 13 |- s e. dom 2nd
40		funfvima 5815	. . . . . . . . . . . . 13 |- ( ( Fun 2nd /\ s e. dom 2nd ) -> ( s e. A -> ( 2nd ` s ) e. ( 2nd " A ) ) )
41	34, 39, 40	mp2an 426	. . . . . . . . . . . 12 |- ( s e. A -> ( 2nd ` s ) e. ( 2nd " A ) )
42	41	ad2antrl 490	. . . . . . . . . . 11 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> ( 2nd ` s ) e. ( 2nd " A ) )
43	42	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> ( 2nd ` s ) e. ( 2nd " A ) )
44	27, 31, 43	rspcdva 2881	. . . . . . . . 9 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> w e. ( 2nd ` s ) )
45		prcunqu 7597	. . . . . . . . 9 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ w e. ( 2nd ` s ) ) -> ( w <Q q -> q e. ( 2nd ` s ) ) )
46	26, 44, 45	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> ( w <Q q -> q e. ( 2nd ` s ) ) )
47	21, 46	mpd 13	. . . . . . 7 |- ( ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q q ) ) -> q e. ( 2nd ` s ) )
48	20, 47	rexlimddv 2627	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. ( 2nd ` s ) )
49	4, 48	jca 306	. . . . 5 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
50		simprl 529	. . . . . . . 8 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. A )
51	10, 50	sseldd 3193	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. P. )
52	51, 25	syl 14	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
53		simpllr 534	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. Q. )
54		prdisj 7604	. . . . . 6 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ q e. Q. ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
55	52, 53, 54	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> -. ( q e. ( 1st ` s ) /\ q e. ( 2nd ` s ) ) )
56	49, 55	pm2.21fal 1392	. . . 4 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> F. )
57	3, 56	rexlimddv 2627	. . 3 |- ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) -> F. )
58	57	inegd 1391	. 2 |- ( ( ph /\ q e. Q. ) -> -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
59	58	ralrimiva 2578	1 |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1372   F. wfal 1377   E.wex 1514   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   _Vcvv 2771   C_ wss 3165   <.cop 3635   U.cuni 3849   |^|cint 3884   class class class wbr 4043   dom cdm 4674   "cima 4677   Fun wfun 5264   -->wf 5266   -onto->wfo 5268   ` cfv 5270   1stc1st 6223   2ndc2nd 6224   Q.cnq 7392   <Q cltq 7397   P.cnp 7403   <P cltp 7407

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-qs 6625  df-ni 7416  df-nqqs 7460  df-ltnqqs 7465  df-inp 7578  df-iltp 7582

This theorem is referenced by:  suplocexprlemex  7834