Theorem zorn2lem5 9322

Description: Lemma for zorn2 9328. (Contributed by NM, 4-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 }
zorn2lem.7	|- H = { z e. A | A. g e. ( F " y ) g R z }

Assertion

Ref	Expression
zorn2lem5	|- ( ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) ) -> ( F " x ) C_ A )

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

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

Proof of Theorem zorn2lem5

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zorn2lem.3	. . . . . 6 |- F = recs ( ( f e. _V |-> ( iota_ v e. C A. u e. C -. u w v ) ) )
2	1	tfr1 7493	. . . . 5 |- F Fn On
3		fnfun 5988	. . . . 5 |- ( F Fn On -> Fun F )
4	2, 3	ax-mp 5	. . . 4 |- Fun F
5		fvelima 6248	. . . 4 |- ( ( Fun F /\ s e. ( F " x ) ) -> E. y e. x ( F ` y ) = s )
6	4, 5	mpan 706	. . 3 |- ( s e. ( F " x ) -> E. y e. x ( F ` y ) = s )
7		nfv 1843	. . . . 5 |- F/ y ( w We A /\ x e. On )
8		nfra1 2941	. . . . 5 |- F/ y A. y e. x H =/= (/)
9	7, 8	nfan 1828	. . . 4 |- F/ y ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) )
10		nfv 1843	. . . 4 |- F/ y s e. A
11		df-ral 2917	. . . . . 6 |- ( A. y e. x H =/= (/) <-> A. y ( y e. x -> H =/= (/) ) )
12		onelon 5748	. . . . . . . . . . . . 13 |- ( ( x e. On /\ y e. x ) -> y e. On )
13		zorn2lem.7	. . . . . . . . . . . . . . . 16 |- H = { z e. A | A. g e. ( F " y ) g R z }
14		ssrab2 3687	. . . . . . . . . . . . . . . 16 |- { z e. A | A. g e. ( F " y ) g R z } C_ A
15	13, 14	eqsstri 3635	. . . . . . . . . . . . . . 15 |- H C_ A
16		zorn2lem.4	. . . . . . . . . . . . . . . 16 |- C = { z e. A | A. g e. ran f g R z }
17	1, 16, 13	zorn2lem1 9318	. . . . . . . . . . . . . . 15 |- ( ( y e. On /\ ( w We A /\ H =/= (/) ) ) -> ( F ` y ) e. H )
18	15, 17	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( y e. On /\ ( w We A /\ H =/= (/) ) ) -> ( F ` y ) e. A )
19		eleq1 2689	. . . . . . . . . . . . . 14 |- ( ( F ` y ) = s -> ( ( F ` y ) e. A <-> s e. A ) )
20	18, 19	syl5ib 234	. . . . . . . . . . . . 13 |- ( ( F ` y ) = s -> ( ( y e. On /\ ( w We A /\ H =/= (/) ) ) -> s e. A ) )
21	12, 20	sylani 686	. . . . . . . . . . . 12 |- ( ( F ` y ) = s -> ( ( ( x e. On /\ y e. x ) /\ ( w We A /\ H =/= (/) ) ) -> s e. A ) )
22	21	com12 32	. . . . . . . . . . 11 |- ( ( ( x e. On /\ y e. x ) /\ ( w We A /\ H =/= (/) ) ) -> ( ( F ` y ) = s -> s e. A ) )
23	22	exp43 640	. . . . . . . . . 10 |- ( x e. On -> ( y e. x -> ( w We A -> ( H =/= (/) -> ( ( F ` y ) = s -> s e. A ) ) ) ) )
24	23	com3r 87	. . . . . . . . 9 |- ( w We A -> ( x e. On -> ( y e. x -> ( H =/= (/) -> ( ( F ` y ) = s -> s e. A ) ) ) ) )
25	24	imp 445	. . . . . . . 8 |- ( ( w We A /\ x e. On ) -> ( y e. x -> ( H =/= (/) -> ( ( F ` y ) = s -> s e. A ) ) ) )
26	25	a2d 29	. . . . . . 7 |- ( ( w We A /\ x e. On ) -> ( ( y e. x -> H =/= (/) ) -> ( y e. x -> ( ( F ` y ) = s -> s e. A ) ) ) )
27	26	spsd 2057	. . . . . 6 |- ( ( w We A /\ x e. On ) -> ( A. y ( y e. x -> H =/= (/) ) -> ( y e. x -> ( ( F ` y ) = s -> s e. A ) ) ) )
28	11, 27	syl5bi 232	. . . . 5 |- ( ( w We A /\ x e. On ) -> ( A. y e. x H =/= (/) -> ( y e. x -> ( ( F ` y ) = s -> s e. A ) ) ) )
29	28	imp 445	. . . 4 |- ( ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) ) -> ( y e. x -> ( ( F ` y ) = s -> s e. A ) ) )
30	9, 10, 29	rexlimd 3026	. . 3 |- ( ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) ) -> ( E. y e. x ( F ` y ) = s -> s e. A ) )
31	6, 30	syl5 34	. 2 |- ( ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) ) -> ( s e. ( F " x ) -> s e. A ) )
32	31	ssrdv 3609	1 |- ( ( ( w We A /\ x e. On ) /\ A. y e. x H =/= (/) ) -> ( F " x ) C_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   We wwe 5072   ran crn 5115   "cima 5117   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:  zorn2lem6  9323  zorn2lem7  9324