Theorem infxpidmlem8 7651

Description: Lemma for infxpidm 7656. The union of a collection of chains C in the collection of bijections H belongs to H. This property will be needed to apply Zorn's Lemma in infxpidmlem9 7652.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem6.2	|- B = ran U. C
infxpidmlem8.3	|- C e. V

Assertion

Ref	Expression
infxpidmlem8	|- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

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

Proof of Theorem infxpidmlem8

Step	Hyp	Ref	Expression
1		ssel2 2115	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> g e. H)
2		infxpidmlem.1	. . . . . . . . . . . . . 14 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3		visset 1860	. . . . . . . . . . . . . 14 |- g e. V
4	2, 3	infxpidmlem2 7645	. . . . . . . . . . . . 13 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
5	4	biimpi 158	. . . . . . . . . . . 12 |- (g e. H -> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
6	5	ord 239	. . . . . . . . . . 11 |- (g e. H -> (-. g = (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
7		f1ofo 3752	. . . . . . . . . . . . . . . . 17 |- (g:(x X. x)-1-1-onto->x -> g:(x X. x)-onto->x)
8		forn 3731	. . . . . . . . . . . . . . . . 17 |- (g:(x X. x)-onto->x -> ran g = x)
9	7, 8	syl 10	. . . . . . . . . . . . . . . 16 |- (g:(x X. x)-1-1-onto->x -> ran g = x)
10	9	eqcomd 1527	. . . . . . . . . . . . . . 15 |- (g:(x X. x)-1-1-onto->x -> x = ran g)
11	10	anim1i 341	. . . . . . . . . . . . . 14 |- ((g:(x X. x)-1-1-onto->x /\ (om ~<_ x /\ x (_ A)) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
12	11	ancoms 447	. . . . . . . . . . . . 13 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
13	12	19.22i 1081	. . . . . . . . . . . 12 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> E.x(x = ran g /\ (om ~<_ x /\ x (_ A)))
14	3	rnex 3418	. . . . . . . . . . . . 13 |- ran g e. V
15		breq2 2678	. . . . . . . . . . . . . 14 |- (x = ran g -> (om ~<_ x <-> om ~<_ ran g))
16		sseq1 2133	. . . . . . . . . . . . . 14 |- (x = ran g -> (x (_ A <-> ran g (_ A))
17	15, 16	anbi12d 639	. . . . . . . . . . . . 13 |- (x = ran g -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ ran g /\ ran g (_ A)))
18	14, 17	ceqsexv 1882	. . . . . . . . . . . 12 |- (E.x(x = ran g /\ (om ~<_ x /\ x (_ A)) <-> (om ~<_ ran g /\ ran g (_ A))
19	13, 18	sylib 205	. . . . . . . . . . 11 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ ran g /\ ran g (_ A))
20	6, 19	syl6 22	. . . . . . . . . 10 |- (g e. H -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
21	1, 20	syl 10	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
22		domtr 4476	. . . . . . . . . . . . 13 |- ((om ~<_ ran g /\ ran g ~<_ B) -> om ~<_ B)
23		ra4e 1742	. . . . . . . . . . . . . . . . 17 |- ((g e. C /\ y e. ran g) -> E.g e. C y e. ran g)
24		infxpidmlem6.2	. . . . . . . . . . . . . . . . . 18 |- B = ran U. C
25	2, 24	infxpidmlem6 7649	. . . . . . . . . . . . . . . . 17 |- (y e. B <-> E.g e. C y e. ran g)
26	23, 25	sylibr 207	. . . . . . . . . . . . . . . 16 |- ((g e. C /\ y e. ran g) -> y e. B)
27	26	ex 380	. . . . . . . . . . . . . . 15 |- (g e. C -> (y e. ran g -> y e. B))
28	27	ssrdv 2121	. . . . . . . . . . . . . 14 |- (g e. C -> ran g (_ B)
29		ssdomg 4469	. . . . . . . . . . . . . . 15 |- (ran g e. V -> (ran g (_ B -> ran g ~<_ B))
30	14, 29	ax-mp 7	. . . . . . . . . . . . . 14 |- (ran g (_ B -> ran g ~<_ B)
31	28, 30	syl 10	. . . . . . . . . . . . 13 |- (g e. C -> ran g ~<_ B)
32	22, 31	sylan2 462	. . . . . . . . . . . 12 |- ((om ~<_ ran g /\ g e. C) -> om ~<_ B)
33	32	expcom 381	. . . . . . . . . . 11 |- (g e. C -> (om ~<_ ran g -> om ~<_ B))
34	33	adantl 397	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> (om ~<_ ran g -> om ~<_ B))
35	34	adantrd 400	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> ((om ~<_ ran g /\ ran g (_ A) -> om ~<_ B))
36	21, 35	syld 27	. . . . . . . 8 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> om ~<_ B))
37	36	r19.23adva 1794	. . . . . . 7 |- (C (_ H -> (E.g e. C -. g = (/) -> om ~<_ B))
38		uni0b 2577	. . . . . . . . . 10 |- (U.C = (/) <-> C (_ {(/)})
39		dfss3 2110	. . . . . . . . . 10 |- (C (_ {(/)} <-> A.g e. C g e. {(/)})
40		elsn 2473	. . . . . . . . . . 11 |- (g e. {(/)} <-> g = (/))
41	40	ralbii 1714	. . . . . . . . . 10 |- (A.g e. C g e. {(/)} <-> A.g e. C g = (/))
42	38, 39, 41	3bitri 184	. . . . . . . . 9 |- (U.C = (/) <-> A.g e. C g = (/))
43	42	notbii 194	. . . . . . . 8 |- (-. U.C = (/) <-> -. A.g e. C g = (/))
44		rexnal 1701	. . . . . . . 8 |- (E.g e. C -. g = (/) <-> -. A.g e. C g = (/))
45	43, 44	bitr4i 183	. . . . . . 7 |- (-. U.C = (/) <-> E.g e. C -. g = (/))
46	37, 45	syl5ib 213	. . . . . 6 |- (C (_ H -> (-. U.C = (/) -> om ~<_ B))
47		pm3.27 330	. . . . . . . . . . . 12 |- ((om ~<_ ran g /\ ran g (_ A) -> ran g (_ A)
48	21, 47	syl6 22	. . . . . . . . . . 11 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> ran g (_ A))
49		rneq 3396	. . . . . . . . . . . . 13 |- (g = (/) -> ran g = ran (/))
50		rn0 3412	. . . . . . . . . . . . 13 |- ran (/) = (/)
51	49, 50	syl6eq 1570	. . . . . . . . . . . 12 |- (g = (/) -> ran g = (/))
52		0ss 2353	. . . . . . . . . . . . 13 |- (/) (_ A
53	52	a1i 8	. . . . . . . . . . . 12 |- (g = (/) -> (/) (_ A)
54	51, 53	eqsstrd 2146	. . . . . . . . . . 11 |- (g = (/) -> ran g (_ A)
55	48, 54	pm2.61d2 135	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> ran g (_ A)
56	55	sseld 2118	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> (y e. ran g -> y e. A))
57	56	r19.23adva 1794	. . . . . . . 8 |- (C (_ H -> (E.g e. C y e. ran g -> y e. A))
58	57, 25	syl5ib 213	. . . . . . 7 |- (C (_ H -> (y e. B -> y e. A))
59	58	ssrdv 2121	. . . . . 6 |- (C (_ H -> B (_ A)
60	46, 59	jctird 613	. . . . 5 |- (C (_ H -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
61	60	adantr 398	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
62	2, 24	infxpidmlem7 7650	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C:(B X. B)-1-1-onto->B)
63	61, 62	jctird 613	. . 3 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> ((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B)))
64		infxpidmlem8.3	. . . . 5 |- C e. V
65	64	uniex 2926	. . . 4 |- U.C e. V
66	65	rnex 3418	. . . . 5 |- ran U. C e. V
67	24, 66	eqeltri 1591	. . . 4 |- B e. V
68	2, 65, 67	infxpidmlem3 7646	. . 3 |- (((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B) -> U.C e. H)
69	63, 68	syl6 22	. 2 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> U.C e. H))
70		orc 276	. . 3 |- (U.C = (/) -> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
71	2, 65	infxpidmlem2 7645	. . 3 |- (U.C e. H <-> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
72	70, 71	sylibr 207	. 2 |- (U.C = (/) -> U.C e. H)
73	69, 72	pm2.61d2 135	1 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 229   /\ wa 230   = wceq 997   e. wcel 999  E.wex 1021  {cab 1509  A.wral 1692  E.wrex 1693  Vcvv 1858   (_ wss 2098  (/)c0 2331  {csn 2461  U.cuni 2557   class class class wbr 2674  omcom 3188   X. cxp 3225  ran crn 3228  -onto->wfo 3237  -1-1-onto->wf1o 3238   ~<_ cdom 4426

This theorem is referenced by:  infxpidmlem9 7652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-iun 2622  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-f1 3252  df-fo 3253  df-f1o 3254  df-en 4429  df-dom 4430

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