Theorem infxpidmlem9 8829

Description: Lemma for infxpidm 8833. By Zorn's Lemma zorn 5959, the collection H (which we show here to be a set) has a maximal element.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem.2	|- A e. _V

Assertion

Ref	Expression
infxpidmlem9	|- E.g e. H A.h e. H -. g C. h

Distinct variable groups:   f,g,h,t,A   g,H,h

Proof of Theorem infxpidmlem9

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . . . 5 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))}
2		unab 2859	. . . . 5 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t)}) = {f | (f = (/) \/ E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))}
3	1, 2	eqtr4i 1911	. . . 4 |- H = ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t)})
4		df-sn 3049	. . . . . 6 |- {(/)} = {f | f = (/)}
5		p0ex 3495	. . . . . 6 |- {(/)} e. _V
6	4, 5	eqeltrri 1968	. . . . 5 |- {f | f = (/)} e. _V
7		df-rex 2110	. . . . . . . 8 |- (E.t e. ~P A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t(t e. ~PA /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
8		visset 2295	. . . . . . . . . . . 12 |- t e. _V
9	8	elpw 3037	. . . . . . . . . . 11 |- (t e. ~PA <-> t C_ A)
10	9	anbi1i 539	. . . . . . . . . 10 |- ((t e. ~PA /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> (t C_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
11		ancom 482	. . . . . . . . . 10 |- ((t C_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t C_ A))
12		an23 543	. . . . . . . . . 10 |- (((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t C_ A) <-> ((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))
13	10, 11, 12	3bitri 194	. . . . . . . . 9 |- ((t e. ~PA /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))
14	13	exbii 1398	. . . . . . . 8 |- (E.t(t e. ~PA /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))
15	7, 14	bitri 190	. . . . . . 7 |- (E.t e. ~P A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t))
16	15	abbii 2006	. . . . . 6 |- {f | E.t e. ~P A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} = {f | E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t)}
17		infxpidmlem.2	. . . . . . . 8 |- A e. _V
18	17	pwex 3487	. . . . . . 7 |- ~PA e. _V
19	8, 8	xpex 4096	. . . . . . . . 9 |- (t X. t) e. _V
20		mapex 5387	. . . . . . . . 9 |- (((t X. t) e. _V /\ t e. _V) -> {f | f:(t X. t)-->t} e. _V)
21	19, 8, 20	mp2an 761	. . . . . . . 8 |- {f | f:(t X. t)-->t} e. _V
22		f1of 4635	. . . . . . . . . 10 |- (f:(t X. t)-1-1-onto->t -> f:(t X. t)-->t)
23	22	adantl 424	. . . . . . . . 9 |- ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) -> f:(t X. t)-->t)
24	23	ss2abi 2679	. . . . . . . 8 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} C_ {f | f:(t X. t)-->t}
25	21, 24	ssexi 3456	. . . . . . 7 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. _V
26	18, 25	abrexex2 4847	. . . . . 6 |- {f | E.t e. ~P A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. _V
27	16, 26	eqeltrri 1968	. . . . 5 |- {f | E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t)} e. _V
28	6, 27	unex 3796	. . . 4 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t C_ A) /\ f:(t X. t)-1-1-onto->t)}) e. _V
29	3, 28	eqeltri 1967	. . 3 |- H e. _V
30	29	zorn 5959	. 2 |- (A.z((z C_ H /\ A.g e. z A.h e. z (g C_ h \/ h C_ g)) -> U.z e. H) -> E.g e. H A.h e. H -. g C. h)
31		eqid 1884	. . 3 |- ran U. z = ran U. z
32		visset 2295	. . 3 |- z e. _V
33	1, 31, 32	infxpidmlem8 8828	. 2 |- ((z C_ H /\ A.g e. z A.h e. z (g C_ h \/ h C_ g)) -> U.z e. H)
34	30, 33	mpg 1332	1 |- E.g e. H A.h e. H -. g C. h

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593   C. wpss 2594  (/)c0 2875  ~Pcpw 3032  {csn 3044  U.cuni 3177   class class class wbr 3338  omcom 3949   X. cxp 3984  ran crn 3987  -->wf 3994  -1-1-onto->wf1o 3997   ~<_ cdom 5424

This theorem is referenced by:  infxpidmlem12 8832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-en 5427  df-dom 5428

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