Theorem zorn2lem4 9321

Description: Lemma for zorn2 9328. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
zorn2lem.3	|- F = recs ( ( f e. _V |-> ( iota_ v e. C A. u e. C -. u w v ) ) )
zorn2lem.4	|- C = { z e. A | A. g e. ran f g R z }
zorn2lem.5	|- D = { z e. A | A. g e. ( F " x ) g R z }

Assertion

Ref	Expression
zorn2lem4	|- ( ( R Po A /\ w We A ) -> E. x e. On D = (/) )

Distinct variable groups:   f, g, u, v, w, x, z, A   D, f, u, v   f, F, g, u, v, x, z   R, f, g, u, v, w, x, z   v, C

Allowed substitution hints:   C( x, z, w, u, f, g)   D( x, z, w, g)   F( w)

Proof of Theorem zorn2lem4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 926	. 2 |- -. ( ran F e. _V /\ -. ran F e. _V )
2		df-ne 2795	. . . . 5 |- ( D =/= (/) <-> -. D = (/) )
3	2	ralbii 2980	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x e. On -. D = (/) )
4		df-ral 2917	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x ( x e. On -> D =/= (/) ) )
5		ralnex 2992	. . . 4 |- ( A. x e. On -. D = (/) <-> -. E. x e. On D = (/) )
6	3, 4, 5	3bitr3i 290	. . 3 |- ( A. x ( x e. On -> D =/= (/) ) <-> -. E. x e. On D = (/) )
7		weso 5105	. . . . . . . . 9 |- ( w We A -> w Or A )
8	7	adantr 481	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> w Or A )
9		vex 3203	. . . . . . . 8 |- w e. _V
10		soex 7109	. . . . . . . 8 |- ( ( w Or A /\ w e. _V ) -> A e. _V )
11	8, 9, 10	sylancl 694	. . . . . . 7 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> A e. _V )
12		zorn2lem.3	. . . . . . . . . . 11 |- F = recs ( ( f e. _V |-> ( iota_ v e. C A. u e. C -. u w v ) ) )
13	12	tfr1 7493	. . . . . . . . . 10 |- F Fn On
14		fvelrnb 6243	. . . . . . . . . 10 |- ( F Fn On -> ( y e. ran F <-> E. x e. On ( F ` x ) = y ) )
15	13, 14	ax-mp 5	. . . . . . . . 9 |- ( y e. ran F <-> E. x e. On ( F ` x ) = y )
16		nfv 1843	. . . . . . . . . . 11 |- F/ x w We A
17		nfa1 2028	. . . . . . . . . . 11 |- F/ x A. x ( x e. On -> D =/= (/) )
18	16, 17	nfan 1828	. . . . . . . . . 10 |- F/ x ( w We A /\ A. x ( x e. On -> D =/= (/) ) )
19		nfv 1843	. . . . . . . . . 10 |- F/ x y e. A
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 |- D = { z e. A | A. g e. ( F " x ) g R z }
21		ssrab2 3687	. . . . . . . . . . . . . . . . . 18 |- { z e. A | A. g e. ( F " x ) g R z } C_ A
22	20, 21	eqsstri 3635	. . . . . . . . . . . . . . . . 17 |- D C_ A
23		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 |- C = { z e. A | A. g e. ran f g R z }
24	12, 23, 20	zorn2lem1 9318	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. D )
25	22, 24	sseldi 3601	. . . . . . . . . . . . . . . 16 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. A )
26		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( ( F ` x ) = y -> ( ( F ` x ) e. A <-> y e. A ) )
27	25, 26	syl5ibcom 235	. . . . . . . . . . . . . . 15 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( ( F ` x ) = y -> y e. A ) )
28	27	exp32 631	. . . . . . . . . . . . . 14 |- ( x e. On -> ( w We A -> ( D =/= (/) -> ( ( F ` x ) = y -> y e. A ) ) ) )
29	28	com12 32	. . . . . . . . . . . . 13 |- ( w We A -> ( x e. On -> ( D =/= (/) -> ( ( F ` x ) = y -> y e. A ) ) ) )
30	29	a2d 29	. . . . . . . . . . . 12 |- ( w We A -> ( ( x e. On -> D =/= (/) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) ) )
31	30	spsd 2057	. . . . . . . . . . 11 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) ) )
32	31	imp 445	. . . . . . . . . 10 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) )
33	18, 19, 32	rexlimd 3026	. . . . . . . . 9 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( E. x e. On ( F ` x ) = y -> y e. A ) )
34	15, 33	syl5bi 232	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( y e. ran F -> y e. A ) )
35	34	ssrdv 3609	. . . . . . 7 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F C_ A )
36	11, 35	ssexd 4805	. . . . . 6 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F e. _V )
37	36	ex 450	. . . . 5 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
38	37	adantl 482	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
39	12, 23, 20	zorn2lem3 9320	. . . . . . . . . . . . . 14 |- ( ( R Po A /\ ( x e. On /\ ( w We A /\ D =/= (/) ) ) ) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) )
40	39	exp45 642	. . . . . . . . . . . . 13 |- ( R Po A -> ( x e. On -> ( w We A -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
41	40	com23 86	. . . . . . . . . . . 12 |- ( R Po A -> ( w We A -> ( x e. On -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
42	41	imp 445	. . . . . . . . . . 11 |- ( ( R Po A /\ w We A ) -> ( x e. On -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) )
43	42	a2d 29	. . . . . . . . . 10 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> ( x e. On -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) )
44	43	imp4a 614	. . . . . . . . 9 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) ) )
45	44	alrimdv 1857	. . . . . . . 8 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> A. y ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) ) )
46	45	alimdv 1845	. . . . . . 7 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> A. x A. y ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) ) )
47		r2al 2939	. . . . . . 7 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) <-> A. x A. y ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) )
48	46, 47	syl6ibr 242	. . . . . 6 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) ) )
49		ssid 3624	. . . . . . . 8 |- On C_ On
50	13	tz7.48lem 7536	. . . . . . . 8 |- ( ( On C_ On /\ A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) ) -> Fun `' ( F |` On ) )
51	49, 50	mpan 706	. . . . . . 7 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' ( F |` On ) )
52		fnrel 5989	. . . . . . . . . . 11 |- ( F Fn On -> Rel F )
53	13, 52	ax-mp 5	. . . . . . . . . 10 |- Rel F
54		fndm 5990	. . . . . . . . . . . 12 |- ( F Fn On -> dom F = On )
55	13, 54	ax-mp 5	. . . . . . . . . . 11 |- dom F = On
56	55	eqimssi 3659	. . . . . . . . . 10 |- dom F C_ On
57		relssres 5437	. . . . . . . . . 10 |- ( ( Rel F /\ dom F C_ On ) -> ( F |` On ) = F )
58	53, 56, 57	mp2an 708	. . . . . . . . 9 |- ( F |` On ) = F
59	58	cnveqi 5297	. . . . . . . 8 |- `' ( F |` On ) = `' F
60	59	funeqi 5909	. . . . . . 7 |- ( Fun `' ( F |` On ) <-> Fun `' F )
61	51, 60	sylib 208	. . . . . 6 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' F )
62	48, 61	syl6 35	. . . . 5 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> Fun `' F ) )
63		onprc 6984	. . . . . 6 |- -. On e. _V
64		funrnex 7133	. . . . . . . 8 |- ( dom `' F e. _V -> ( Fun `' F -> ran `' F e. _V ) )
65	64	com12 32	. . . . . . 7 |- ( Fun `' F -> ( dom `' F e. _V -> ran `' F e. _V ) )
66		df-rn 5125	. . . . . . . 8 |- ran F = dom `' F
67	66	eleq1i 2692	. . . . . . 7 |- ( ran F e. _V <-> dom `' F e. _V )
68		dfdm4 5316	. . . . . . . . 9 |- dom F = ran `' F
69	55, 68	eqtr3i 2646	. . . . . . . 8 |- On = ran `' F
70	69	eleq1i 2692	. . . . . . 7 |- ( On e. _V <-> ran `' F e. _V )
71	65, 67, 70	3imtr4g 285	. . . . . 6 |- ( Fun `' F -> ( ran F e. _V -> On e. _V ) )
72	63, 71	mtoi 190	. . . . 5 |- ( Fun `' F -> -. ran F e. _V )
73	62, 72	syl6 35	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> -. ran F e. _V ) )
74	38, 73	jcad 555	. . 3 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> ( ran F e. _V /\ -. ran F e. _V ) ) )
75	6, 74	syl5bir 233	. 2 |- ( ( R Po A /\ w We A ) -> ( -. E. x e. On D = (/) -> ( ran F e. _V /\ -. ran F e. _V ) ) )
76	1, 75	mt3i 141	1 |- ( ( R Po A /\ w We A ) -> E. x e. On D = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   Po wpo 5033   Or wor 5034   We wwe 5072   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   iota_crio 6610  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  zorn2lem7  9324