Theorem zorn2lem2 9319

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
zorn2lem2	|- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( y e. x -> ( F ` y ) R ( F ` x ) ) )

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

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

Proof of Theorem zorn2lem2

Step	Hyp	Ref	Expression
1		zorn2lem.3	. . . 4 |- F = recs ( ( f e. _V |-> ( iota_ v e. C A. u e. C -. u w v ) ) )
2		zorn2lem.4	. . . 4 |- C = { z e. A | A. g e. ran f g R z }
3		zorn2lem.5	. . . 4 |- D = { z e. A | A. g e. ( F " x ) g R z }
4	1, 2, 3	zorn2lem1 9318	. . 3 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( F ` x ) e. D )
5		breq2 4657	. . . . . 6 |- ( z = ( F ` x ) -> ( g R z <-> g R ( F ` x ) ) )
6	5	ralbidv 2986	. . . . 5 |- ( z = ( F ` x ) -> ( A. g e. ( F " x ) g R z <-> A. g e. ( F " x ) g R ( F ` x ) ) )
7	6, 3	elrab2 3366	. . . 4 |- ( ( F ` x ) e. D <-> ( ( F ` x ) e. A /\ A. g e. ( F " x ) g R ( F ` x ) ) )
8	7	simprbi 480	. . 3 |- ( ( F ` x ) e. D -> A. g e. ( F " x ) g R ( F ` x ) )
9	4, 8	syl 17	. 2 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> A. g e. ( F " x ) g R ( F ` x ) )
10	1	tfr1 7493	. . . 4 |- F Fn On
11		onss 6990	. . . 4 |- ( x e. On -> x C_ On )
12		fnfvima 6496	. . . . 5 |- ( ( F Fn On /\ x C_ On /\ y e. x ) -> ( F ` y ) e. ( F " x ) )
13	12	3expia 1267	. . . 4 |- ( ( F Fn On /\ x C_ On ) -> ( y e. x -> ( F ` y ) e. ( F " x ) ) )
14	10, 11, 13	sylancr 695	. . 3 |- ( x e. On -> ( y e. x -> ( F ` y ) e. ( F " x ) ) )
15	14	adantr 481	. 2 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( y e. x -> ( F ` y ) e. ( F " x ) ) )
16		breq1 4656	. . 3 |- ( g = ( F ` y ) -> ( g R ( F ` x ) <-> ( F ` y ) R ( F ` x ) ) )
17	16	rspccv 3306	. 2 |- ( A. g e. ( F " x ) g R ( F ` x ) -> ( ( F ` y ) e. ( F " x ) -> ( F ` y ) R ( F ` x ) ) )
18	9, 15, 17	sylsyld 61	1 |- ( ( x e. On /\ ( w We A /\ D =/= (/) ) ) -> ( y e. x -> ( F ` y ) R ( F ` x ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {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   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:  zorn2lem3  9320  zorn2lem6  9323