Theorem zorn2lem4 8926

Description: Lemma for zorn2 8933. (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		pm3.24 892	. 2 |- -. ( ran F e. _V /\ -. ran F e. _V )
2		df-ne 2623	. . . . 5 |- ( D =/= (/) <-> -. D = (/) )
3	2	ralbii 2818	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x e. On -. D = (/) )
4		df-ral 2741	. . . 4 |- ( A. x e. On D =/= (/) <-> A. x ( x e. On -> D =/= (/) ) )
5		ralnex 2833	. . . 4 |- ( A. x e. On -. D = (/) <-> -. E. x e. On D = (/) )
6	3, 4, 5	3bitr3i 279	. . 3 |- ( A. x ( x e. On -> D =/= (/) ) <-> -. E. x e. On D = (/) )
7		weso 4824	. . . . . . . . 9 |- ( w We A -> w Or A )
8	7	adantr 467	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> w Or A )
9		vex 3047	. . . . . . . 8 |- w e. _V
10		soex 6733	. . . . . . . 8 |- ( ( w Or A /\ w e. _V ) -> A e. _V )
11	8, 9, 10	sylancl 667	. . . . . . 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 7112	. . . . . . . . . 10 |- F Fn On
14		fvelrnb 5910	. . . . . . . . . 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 1760	. . . . . . . . . . 11 |- F/ x w We A
17		nfa1 1978	. . . . . . . . . . 11 |- F/ x A. x ( x e. On -> D =/= (/) )
18	16, 17	nfan 2010	. . . . . . . . . 10 |- F/ x ( w We A /\ A. x ( x e. On -> D =/= (/) ) )
19		nfv 1760	. . . . . . . . . 10 |- F/ x y e. A
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 |- D = { z e. A | A. g e. ( F " x ) g R z }
21		ssrab2 3513	. . . . . . . . . . . . . . . . . 18 |- { z e. A | A. g e. ( F " x ) g R z } C_ A
22	20, 21	eqsstri 3461	. . . . . . . . . . . . . . . . 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 8923	. . . . . . . . . . . . . . . . 17 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. D )
25	22, 24	sseldi 3429	. . . . . . . . . . . . . . . 16 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. A )
26		eleq1 2516	. . . . . . . . . . . . . . . 16 |- ( ( F ` x ) = y -> ( ( F ` x ) e. A <-> y e. A ) )
27	25, 26	syl5ibcom 224	. . . . . . . . . . . . . . 15 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( ( F ` x ) = y -> y e. A ) )
28	27	exp32 609	. . . . . . . . . . . . . 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 1944	. . . . . . . . . . 11 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) ) )
32	31	imp 431	. . . . . . . . . 10 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( x e. On -> ( ( F ` x ) = y -> y e. A ) ) )
33	18, 19, 32	rexlimd 2870	. . . . . . . . 9 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( E. x e. On ( F ` x ) = y -> y e. A ) )
34	15, 33	syl5bi 221	. . . . . . . 8 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ( y e. ran F -> y e. A ) )
35	34	ssrdv 3437	. . . . . . 7 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F C_ A )
36	11, 35	ssexd 4549	. . . . . 6 |- ( ( w We A /\ A. x ( x e. On -> D =/= (/) ) ) -> ran F e. _V )
37	36	ex 436	. . . . 5 |- ( w We A -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
38	37	adantl 468	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> ran F e. _V ) )
39	12, 23, 20	zorn2lem3 8925	. . . . . . . . . . . . . 14 |- ( ( R Po A /\ ( x e. On /\ ( w We A /\ D =/= (/) ) ) ) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) )
40	39	exp45 618	. . . . . . . . . . . . 13 |- ( R Po A -> ( x e. On -> ( w We A -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
41	40	com23 81	. . . . . . . . . . . 12 |- ( R Po A -> ( w We A -> ( x e. On -> ( D =/= (/) -> ( y e. x -> -. ( F ` x ) = ( F ` y ) ) ) ) ) )
42	41	imp 431	. . . . . . . . . . 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 593	. . . . . . . . 9 |- ( ( R Po A /\ w We A ) -> ( ( x e. On -> D =/= (/) ) -> ( ( x e. On /\ y e. x ) -> -. ( F ` x ) = ( F ` y ) ) ) )
45	44	alrimdv 1774	. . . . . . . 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 1762	. . . . . . 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 2765	. . . . . . 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 231	. . . . . 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 3450	. . . . . . . 8 |- On C_ On
50	13	tz7.48lem 7155	. . . . . . . 8 |- ( ( On C_ On /\ A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) ) -> Fun `' ( F |` On ) )
51	49, 50	mpan 675	. . . . . . 7 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' ( F |` On ) )
52		fnrel 5672	. . . . . . . . . . 11 |- ( F Fn On -> Rel F )
53	13, 52	ax-mp 5	. . . . . . . . . 10 |- Rel F
54		fndm 5673	. . . . . . . . . . . 12 |- ( F Fn On -> dom F = On )
55	13, 54	ax-mp 5	. . . . . . . . . . 11 |- dom F = On
56	55	eqimssi 3485	. . . . . . . . . 10 |- dom F C_ On
57		relssres 5141	. . . . . . . . . 10 |- ( ( Rel F /\ dom F C_ On ) -> ( F |` On ) = F )
58	53, 56, 57	mp2an 677	. . . . . . . . 9 |- ( F |` On ) = F
59	58	cnveqi 5008	. . . . . . . 8 |- `' ( F |` On ) = `' F
60	59	funeqi 5601	. . . . . . 7 |- ( Fun `' ( F |` On ) <-> Fun `' F )
61	51, 60	sylib 200	. . . . . 6 |- ( A. x e. On A. y e. x -. ( F ` x ) = ( F ` y ) -> Fun `' F )
62	48, 61	syl6 34	. . . . 5 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> Fun `' F ) )
63		onprc 6608	. . . . . 6 |- -. On e. _V
64		funrnex 6757	. . . . . . . 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 4844	. . . . . . . 8 |- ran F = dom `' F
67	66	eleq1i 2519	. . . . . . 7 |- ( ran F e. _V <-> dom `' F e. _V )
68		dfdm4 5026	. . . . . . . . 9 |- dom F = ran `' F
69	55, 68	eqtr3i 2474	. . . . . . . 8 |- On = ran `' F
70	69	eleq1i 2519	. . . . . . 7 |- ( On e. _V <-> ran `' F e. _V )
71	65, 67, 70	3imtr4g 274	. . . . . 6 |- ( Fun `' F -> ( ran F e. _V -> On e. _V ) )
72	63, 71	mtoi 182	. . . . 5 |- ( Fun `' F -> -. ran F e. _V )
73	62, 72	syl6 34	. . . 4 |- ( ( R Po A /\ w We A ) -> ( A. x ( x e. On -> D =/= (/) ) -> -. ran F e. _V ) )
74	38, 73	jcad 536	. . 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 222	. 2 |- ( ( R Po A /\ w We A ) -> ( -. E. x e. On D = (/) -> ( ran F e. _V /\ -. ran F e. _V ) ) )
76	1, 75	mt3i 130	1 |- ( ( R Po A /\ w We A ) -> E. x e. On D = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1441   = wceq 1443   e. wcel 1886   =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044   C_ wss 3403   (/)c0 3730   class class class wbr 4401   |-> cmpt 4460   Po wpo 4752   Or wor 4753   We wwe 4791   `'ccnv 4832   dom cdm 4833   ran crn 4834   |` cres 4835   "cima 4836   Rel wrel 4838   Oncon0 5422   Fun wfun 5575   Fn wfn 5576   ` cfv 5581   iota_crio 6249  recscrecs 7086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-wrecs 7025  df-recs 7087

This theorem is referenced by:  zorn2lem7  8929