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Theorem zorn2lem1 8657
Description: Lemma for zorn2 8667. (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
zorn2lem1  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  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 zorn2lem1
StepHypRef Expression
1 zorn2lem.3 . . . . 5  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
21tfr2 6849 . . . 4  |-  ( x  e.  On  ->  ( F `  x )  =  ( ( f  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) `
 ( F  |`  x ) ) )
32adantr 465 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  =  ( ( f  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) `  ( F  |`  x ) ) )
41tfr1 6848 . . . . . 6  |-  F  Fn  On
5 fnfun 5503 . . . . . 6  |-  ( F  Fn  On  ->  Fun  F )
64, 5ax-mp 5 . . . . 5  |-  Fun  F
7 vex 2970 . . . . 5  |-  x  e. 
_V
8 resfunexg 5938 . . . . 5  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
96, 7, 8mp2an 672 . . . 4  |-  ( F  |`  x )  e.  _V
10 rneq 5060 . . . . . . . . . . . 12  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
11 df-ima 4848 . . . . . . . . . . . 12  |-  ( F
" x )  =  ran  ( F  |`  x )
1210, 11syl6eqr 2488 . . . . . . . . . . 11  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
1312eleq2d 2505 . . . . . . . . . 10  |-  ( f  =  ( F  |`  x )  ->  (
g  e.  ran  f  <->  g  e.  ( F "
x ) ) )
1413imbi1d 317 . . . . . . . . 9  |-  ( f  =  ( F  |`  x )  ->  (
( g  e.  ran  f  ->  g R z )  <->  ( g  e.  ( F " x
)  ->  g R
z ) ) )
1514ralbidv2 2732 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  ( A. g  e.  ran  f  g R z  <->  A. g  e.  ( F " x ) g R z ) )
1615rabbidv 2959 . . . . . . 7  |-  ( f  =  ( F  |`  x )  ->  { z  e.  A  |  A. g  e.  ran  f  g R z }  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z } )
17 zorn2lem.4 . . . . . . 7  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
18 zorn2lem.5 . . . . . . 7  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
1916, 17, 183eqtr4g 2495 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  C  =  D )
2019eleq2d 2505 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  (
u  e.  C  <->  u  e.  D ) )
2120imbi1d 317 . . . . . . 7  |-  ( f  =  ( F  |`  x )  ->  (
( u  e.  C  ->  -.  u w v )  <->  ( u  e.  D  ->  -.  u w v ) ) )
2221ralbidv2 2732 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  ( A. u  e.  C  -.  u w v  <->  A. u  e.  D  -.  u w v ) )
2319, 22riotaeqbidv 6050 . . . . 5  |-  ( f  =  ( F  |`  x )  ->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v )  =  ( iota_ v  e.  D  A. u  e.  D  -.  u w v ) )
24 eqid 2438 . . . . 5  |-  ( f  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) )  =  ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) )
25 riotaex 6051 . . . . 5  |-  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e. 
_V
2623, 24, 25fvmpt 5769 . . . 4  |-  ( ( F  |`  x )  e.  _V  ->  ( (
f  e.  _V  |->  (
iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) `  ( F  |`  x ) )  =  ( iota_ v  e.  D  A. u  e.  D  -.  u w v ) )
279, 26ax-mp 5 . . 3  |-  ( ( f  e.  _V  |->  (
iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) `  ( F  |`  x ) )  =  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )
283, 27syl6eq 2486 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  =  (
iota_ v  e.  D  A. u  e.  D  -.  u w v ) )
29 simprl 755 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  We  A
)
30 ssrab2 3432 . . . . . 6  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
3118, 30eqsstri 3381 . . . . 5  |-  D  C_  A
32 weso 4706 . . . . . . 7  |-  ( w  We  A  ->  w  Or  A )
3332ad2antrl 727 . . . . . 6  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  Or  A
)
34 vex 2970 . . . . . 6  |-  w  e. 
_V
35 soex 6516 . . . . . 6  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
3633, 34, 35sylancl 662 . . . . 5  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  A  e.  _V )
37 ssexg 4433 . . . . 5  |-  ( ( D  C_  A  /\  A  e.  _V )  ->  D  e.  _V )
3831, 36, 37sylancr 663 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  e.  _V )
3931a1i 11 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  C_  A
)
40 simprr 756 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  =/=  (/) )
41 wereu 4711 . . . 4  |-  ( ( w  We  A  /\  ( D  e.  _V  /\  D  C_  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
4229, 38, 39, 40, 41syl13anc 1220 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
43 riotacl 6062 . . 3  |-  ( E! v  e.  D  A. u  e.  D  -.  u w v  -> 
( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e.  D )
4442, 43syl 16 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e.  D
)
4528, 44eqeltrd 2512 1  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E!wreu 2712   {crab 2714   _Vcvv 2967    C_ wss 3323   (/)c0 3632   class class class wbr 4287    e. cmpt 4345    Or wor 4635    We wwe 4673   Oncon0 4714   ran crn 4836    |` cres 4837   "cima 4838   Fun wfun 5407    Fn wfn 5408   ` cfv 5413   iota_crio 6046  recscrecs 6823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-recs 6824
This theorem is referenced by:  zorn2lem2  8658  zorn2lem3  8659  zorn2lem4  8660  zorn2lem5  8661
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