Theorem zorn2lem4 8870

Description: Lemma for zorn2 8877. (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		pm3.24 880	. 2 |- -. ( ran F e. _V /\ -. ran F e. _V )
2		df-ne 2651	. . . . 5 |- ( D =/= (/) <-> -. D = (/) )
3	2	ralbii 2885	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x e. On -. D = (/) )
4		df-ral 2809	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x ( x e. On -> D =/= (/) ) )
5		ralnex 2900	. . . 4 |- ( A. x e. On -. D = (/) <-> -. E. x e. On D = (/) )
6	3, 4, 5	3bitr3i 275	. . 3 |- ( A. x ( x e. On -> D =/= (/) ) <-> -. E. x e. On D = (/) )
7		weso 4859	. . . . . . . . 9 |- ( w We A -> w Or A )
8	7	adantr 463	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> w Or A )
9		vex 3109	. . . . . . . 8 |- w e. _V
10		soex 6716	. . . . . . . 8 |- ( ( w Or A /\ w e. _V ) -> A e. _V )
11	8, 9, 10	sylancl 660	. . . . . . 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 7058	. . . . . . . . . 10 |- F Fn On
14		fvelrnb 5895	. . . . . . . . . 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 1712	. . . . . . . . . . 11 |- F/ x w We A
17		nfa1 1902	. . . . . . . . . . 11 |- F/ x A. x ( x e. On -> D =/= (/) )
18	16, 17	nfan 1933	. . . . . . . . . 10 |- F/ x ( w We A /\ A. x ( x e. On -> D =/= (/) ) )
19		nfv 1712	. . . . . . . . . 10 |- F/ x y e. A
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 |- D = { z e. A | A. g e. ( F " x ) g R z }
21		ssrab2 3571	. . . . . . . . . . . . . . . . . 18 |- { z e. A | A. g e. ( F " x ) g R z } C_ A
22	20, 21	eqsstri 3519	. . . . . . . . . . . . . . . . 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 8867	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. D )
25	22, 24	sseldi 3487	. . . . . . . . . . . . . . . 16 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. A )
26		eleq1 2526	. . . . . . . . . . . . . . . 16 |- ( ( F ` x ) = y -> ( ( F ` x ) e. A <-> y e. A ) )
27	25, 26	syl5ibcom 220	. . . . . . . . . . . . . . 15 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( ( F ` x ) = y -> y e. A ) )
28	27	exp32 603	. . . . . . . . . . . . . 14 |- ( x e. On -> ( w We A -> ( D =/= (/) -> ( ( F ` x ) = y -> y e. A ) ) ) )
29	28	com12 31	. . . . . . . . . . . . 13 |- ( w We A -> ( x e. On -> ( D =/= (/) -> ( ( F ` x ) = y -> y e. A ) ) ) )
30	29	a2d 26	. . . . . . . . . . . 12 |- ( w We A -> ( ( x e. On -> D =/= (/) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) ) )
31	30	spsd 1872	. . . . . . . . . . 11 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) ) )
32	31	imp 427	. . . . . . . . . 10 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) )
33	18, 19, 32	rexlimd 2938	. . . . . . . . 9 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( E. x e. On ( F ` x ) = y -> y e. A ) )
34	15, 33	syl5bi 217	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( y e. ran F -> y e. A ) )
35	34	ssrdv 3495	. . . . . . 7 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F C_ A )
36	11, 35	ssexd 4584	. . . . . 6 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F e. _V )
37	36	ex 432	. . . . 5 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
38	37	adantl 464	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
39	12, 23, 20	zorn2lem3 8869	. . . . . . . . . . . . . 14 |- ( ( R Po A /\ ( x e. On /\ ( w We A /\ D =/= (/) ) ) ) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) )
40	39	exp45 612	. . . . . . . . . . . . 13 |- ( R Po A -> ( x e. On -> ( w We A -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
41	40	com23 78	. . . . . . . . . . . 12 |- ( R Po A -> ( w We A -> ( x e. On -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
42	41	imp 427	. . . . . . . . . . 11 |- ( ( R Po A /\ w We A ) -> ( x e. On -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) )
43	42	a2d 26	. . . . . . . . . 10 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> ( x e. On -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) )
44	43	imp4a 587	. . . . . . . . 9 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) ) )
45	44	alrimdv 1726	. . . . . . . 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 1714	. . . . . . 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 2832	. . . . . . 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 227	. . . . . 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 3508	. . . . . . . 8 |- On C_ On
50	13	tz7.48lem 7098	. . . . . . . 8 |- ( ( On C_ On /\ A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) ) -> Fun `' ( F |` On ) )
51	49, 50	mpan 668	. . . . . . 7 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' ( F |` On ) )
52		fnrel 5661	. . . . . . . . . . 11 |- ( F Fn On -> Rel F )
53	13, 52	ax-mp 5	. . . . . . . . . 10 |- Rel F
54		fndm 5662	. . . . . . . . . . . 12 |- ( F Fn On -> dom F = On )
55	13, 54	ax-mp 5	. . . . . . . . . . 11 |- dom F = On
56	55	eqimssi 3543	. . . . . . . . . 10 |- dom F C_ On
57		relssres 5299	. . . . . . . . . 10 |- ( ( Rel F /\ dom F C_ On ) -> ( F |` On ) = F )
58	53, 56, 57	mp2an 670	. . . . . . . . 9 |- ( F |` On ) = F
59	58	cnveqi 5166	. . . . . . . 8 |- `' ( F |` On ) = `' F
60	59	funeqi 5590	. . . . . . 7 |- ( Fun `' ( F |` On ) <-> Fun `' F )
61	51, 60	sylib 196	. . . . . 6 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' F )
62	48, 61	syl6 33	. . . . 5 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> Fun `' F ) )
63		onprc 6593	. . . . . 6 |- -. On e. _V
64		funrnex 6740	. . . . . . . 8 |- ( dom `' F e. _V -> ( Fun `' F -> ran `' F e. _V ) )
65	64	com12 31	. . . . . . 7 |- ( Fun `' F -> ( dom `' F e. _V -> ran `' F e. _V ) )
66		df-rn 4999	. . . . . . . 8 |- ran F = dom `' F
67	66	eleq1i 2531	. . . . . . 7 |- ( ran F e. _V <-> dom `' F e. _V )
68		dfdm4 5184	. . . . . . . . 9 |- dom F = ran `' F
69	55, 68	eqtr3i 2485	. . . . . . . 8 |- On = ran `' F
70	69	eleq1i 2531	. . . . . . 7 |- ( On e. _V <-> ran `' F e. _V )
71	65, 67, 70	3imtr4g 270	. . . . . 6 |- ( Fun `' F -> ( ran F e. _V -> On e. _V ) )
72	63, 71	mtoi 178	. . . . 5 |- ( Fun `' F -> -. ran F e. _V )
73	62, 72	syl6 33	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> -. ran F e. _V ) )
74	38, 73	jcad 531	. . 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 218	. 2 |- ( ( R Po A /\ w We A ) -> ( -. E. x e. On D = (/) -> ( ran F e. _V /\ -. ran F e. _V ) ) )
76	1, 75	mt3i 126	1 |- ( ( R Po A /\ w We A ) -> E. x e. On D = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1396   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   C_ wss 3461   (/)c0 3783   class class class wbr 4439   |-> cmpt 4497   Po wpo 4787   Or wor 4788   We wwe 4826   Oncon0 4867   `'ccnv 4987   dom cdm 4988   ran crn 4989   |` cres 4990   "cima 4991   Rel wrel 4993   Fun wfun 5564   Fn wfn 5565   ` cfv 5570   iota_crio 6231  recscrecs 7033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-recs 7034

This theorem is referenced by:  zorn2lem7  8873