Theorem suplocexprlemdisj 8035

Description: Lemma for suplocexpr 8040. 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 531	. . . . 5 |- ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) -> q e. U. ( 1st " A ) )
2		suplocexprlemell 8028	. . . . 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 533	. . . . . 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 538	. . . . . . . . 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 8030	. . . . . . . . . . . 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 8029	. . . . . . . . . . . 12 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
13	12	eleq2d 2302	. . . . . . . . . . 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 4113	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
16	15	rexbidv 2543	. . . . . . . . . . 11 |- ( u = q -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q q ) )
17	16	elrab 2973	. . . . . . . . . 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 533	. . . . . . . 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 537	. . . . . . . . . . 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 3239	. . . . . . . . . 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 7790	. . . . . . . . . 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 2296	. . . . . . . . . 10 |- ( t = ( 2nd ` s ) -> ( w e. t <-> w e. ( 2nd ` s ) ) )
28		simprl 531	. . . . . . . . . . 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 2816	. . . . . . . . . . . 12 |- w e. _V
30	29	elint2 3956	. . . . . . . . . . 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 6352	. . . . . . . . . . . . . 14 |- 2nd : _V -onto-> _V
33		fofun 5591	. . . . . . . . . . . . . 14 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 |- Fun 2nd
35		vex 2816	. . . . . . . . . . . . . 14 |- s e. _V
36		fof 5590	. . . . . . . . . . . . . . . 16 |- ( 2nd : _V -onto-> _V -> 2nd : _V --> _V )
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 |- 2nd : _V --> _V
38	37	fdmi 5516	. . . . . . . . . . . . . 14 |- dom 2nd = _V
39	35, 38	eleqtrri 2308	. . . . . . . . . . . . 13 |- s e. dom 2nd
40		funfvima 5918	. . . . . . . . . . . . 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 2926	. . . . . . . . 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 7800	. . . . . . . . 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 2665	. . . . . 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 531	. . . . . . . 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 3239	. . . . . . 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 536	. . . . . 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 7807	. . . . . 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 1418	. . . 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 2665	. . 3 |- ( ( ( ph /\ q e. Q. ) /\ ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) ) -> F. )
58	57	inegd 1417	. 2 |- ( ( ph /\ q e. Q. ) -> -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
59	58	ralrimiva 2615	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 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-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  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-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-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-id 4414  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-1st 6334  df-2nd 6335  df-qs 6773  df-ni 7619  df-nqqs 7663  df-ltnqqs 7668  df-inp 7781  df-iltp 7785

This theorem is referenced by:  suplocexprlemex  8037