Theorem suplocexprlemlub 7943

Description: Lemma for suplocexpr 7944. The putative supremum is a least 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
suplocexprlemlub	|- ( ph -> ( y <P B -> E. z e. A y <P z ) )

Distinct variable groups:   y, A, z   x, A, y   z, B   ph, y, z   ph, x

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

Proof of Theorem suplocexprlemlub

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ y <P B ) -> y <P B )
2		ltrelpr 7724	. . . . . . . 8 |- <P C_ ( P. X. P. )
3	2	brel 4778	. . . . . . 7 |- ( y <P B -> ( y e. P. /\ B e. P. ) )
4	3	simpld 112	. . . . . 6 |- ( y <P B -> y e. P. )
5	4	adantl 277	. . . . 5 |- ( ( ph /\ y <P B ) -> y e. P. )
6	3	simprd 114	. . . . . 6 |- ( y <P B -> B e. P. )
7	6	adantl 277	. . . . 5 |- ( ( ph /\ y <P B ) -> B e. P. )
8		ltdfpr 7725	. . . . 5 |- ( ( y e. P. /\ B e. P. ) -> ( y <P B <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) )
9	5, 7, 8	syl2anc 411	. . . 4 |- ( ( ph /\ y <P B ) -> ( y <P B <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) )
10	1, 9	mpbid 147	. . 3 |- ( ( ph /\ y <P B ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) )
11		simprrr 542	. . . . . 6 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 1st ` B ) )
12		suplocexpr.b	. . . . . . . . . 10 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
13	12	fveq2i 5642	. . . . . . . . 9 |- ( 1st ` B ) = ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
14		npex 7692	. . . . . . . . . . . . 13 |- P. e. _V
15	14	a1i 9	. . . . . . . . . . . 12 |- ( ph -> P. e. _V )
16		suplocexpr.m	. . . . . . . . . . . . 13 |- ( ph -> E. x x e. A )
17		suplocexpr.ub	. . . . . . . . . . . . 13 |- ( ph -> E. x e. P. A. y e. A y <P x )
18		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 ) ) )
19	16, 17, 18	suplocexprlemss 7934	. . . . . . . . . . . 12 |- ( ph -> A C_ P. )
20	15, 19	ssexd 4229	. . . . . . . . . . 11 |- ( ph -> A e. _V )
21		fo1st 6319	. . . . . . . . . . . . 13 |- 1st : _V -onto-> _V
22		fofun 5560	. . . . . . . . . . . . 13 |- ( 1st : _V -onto-> _V -> Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 |- Fun 1st
24		funimaexg 5414	. . . . . . . . . . . 12 |- ( ( Fun 1st /\ A e. _V ) -> ( 1st " A ) e. _V )
25	23, 24	mpan 424	. . . . . . . . . . 11 |- ( A e. _V -> ( 1st " A ) e. _V )
26		uniexg 4536	. . . . . . . . . . 11 |- ( ( 1st " A ) e. _V -> U. ( 1st " A ) e. _V )
27	20, 25, 26	3syl 17	. . . . . . . . . 10 |- ( ph -> U. ( 1st " A ) e. _V )
28		nqex 7582	. . . . . . . . . . 11 |- Q. e. _V
29	28	rabex 4234	. . . . . . . . . 10 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V
30		op1stg 6312	. . . . . . . . . 10 |- ( ( U. ( 1st " A ) e. _V /\ { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } e. _V ) -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
31	27, 29, 30	sylancl 413	. . . . . . . . 9 |- ( ph -> ( 1st ` <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. ) = U. ( 1st " A ) )
32	13, 31	eqtrid 2276	. . . . . . . 8 |- ( ph -> ( 1st ` B ) = U. ( 1st " A ) )
33	32	eleq2d 2301	. . . . . . 7 |- ( ph -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
34	33	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> ( s e. ( 1st ` B ) <-> s e. U. ( 1st " A ) ) )
35	11, 34	mpbid 147	. . . . 5 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. U. ( 1st " A ) )
36		suplocexprlemell 7932	. . . . 5 |- ( s e. U. ( 1st " A ) <-> E. z e. A s e. ( 1st ` z ) )
37	35, 36	sylib 122	. . . 4 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> E. z e. A s e. ( 1st ` z ) )
38		simprl 531	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. Q. )
39	38	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. Q. )
40		simprrl 541	. . . . . . . . 9 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> s e. ( 2nd ` y ) )
41	40	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. ( 2nd ` y ) )
42		simpr 110	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> s e. ( 1st ` z ) )
43		rspe 2581	. . . . . . . 8 |- ( ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) )
44	39, 41, 42, 43	syl12anc 1271	. . . . . . 7 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) )
45	4	ad4antlr 495	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> y e. P. )
46	19	ad4antr 494	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> A C_ P. )
47		simplr 529	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> z e. A )
48	46, 47	sseldd 3228	. . . . . . . 8 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> z e. P. )
49		ltdfpr 7725	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( y <P z <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) )
50	45, 48, 49	syl2anc 411	. . . . . . 7 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> ( y <P z <-> E. s e. Q. ( s e. ( 2nd ` y ) /\ s e. ( 1st ` z ) ) ) )
51	44, 50	mpbird 167	. . . . . 6 |- ( ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) /\ s e. ( 1st ` z ) ) -> y <P z )
52	51	ex 115	. . . . 5 |- ( ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) /\ z e. A ) -> ( s e. ( 1st ` z ) -> y <P z ) )
53	52	reximdva 2634	. . . 4 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> ( E. z e. A s e. ( 1st ` z ) -> E. z e. A y <P z ) )
54	37, 53	mpd 13	. . 3 |- ( ( ( ph /\ y <P B ) /\ ( s e. Q. /\ ( s e. ( 2nd ` y ) /\ s e. ( 1st ` B ) ) ) ) -> E. z e. A y <P z )
55	10, 54	rexlimddv 2655	. 2 |- ( ( ph /\ y <P B ) -> E. z e. A y <P z )
56	55	ex 115	1 |- ( ph -> ( y <P B -> E. z e. A y <P z ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   {crab 2514   _Vcvv 2802   C_ wss 3200   <.cop 3672   U.cuni 3893   |^|cint 3928   class class class wbr 4088   "cima 4728   Fun wfun 5320   -onto->wfo 5324   ` cfv 5326   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   <Q cltq 7504   P.cnp 7510   <P cltp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-qs 6707  df-ni 7523  df-nqqs 7567  df-inp 7685  df-iltp 7689

This theorem is referenced by:  suplocexpr  7944