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Theorem zorn2lem1 8874
Description: Lemma for zorn2 8884. (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 7065 . . . 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 7064 . . . . . 6  |-  F  Fn  On
5 fnfun 5664 . . . . . 6  |-  ( F  Fn  On  ->  Fun  F )
64, 5ax-mp 5 . . . . 5  |-  Fun  F
7 vex 3096 . . . . 5  |-  x  e. 
_V
8 resfunexg 6118 . . . . 5  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
96, 7, 8mp2an 672 . . . 4  |-  ( F  |`  x )  e.  _V
10 rneq 5214 . . . . . . . . . . . 12  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
11 df-ima 4998 . . . . . . . . . . . 12  |-  ( F
" x )  =  ran  ( F  |`  x )
1210, 11syl6eqr 2500 . . . . . . . . . . 11  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
1312eleq2d 2511 . . . . . . . . . 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 2876 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  ( A. g  e.  ran  f  g R z  <->  A. g  e.  ( F " x ) g R z ) )
1615rabbidv 3085 . . . . . . 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 2507 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  C  =  D )
2019eleq2d 2511 . . . . . . . 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 2876 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  ( A. u  e.  C  -.  u w v  <->  A. u  e.  D  -.  u w v ) )
2319, 22riotaeqbidv 6241 . . . . 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 2441 . . . . 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 6242 . . . . 5  |-  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e. 
_V
2623, 24, 25fvmpt 5937 . . . 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 2498 . 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 weso 4856 . . . . . . 7  |-  ( w  We  A  ->  w  Or  A )
3130ad2antrl 727 . . . . . 6  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  Or  A
)
32 vex 3096 . . . . . 6  |-  w  e. 
_V
33 soex 6724 . . . . . 6  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
3431, 32, 33sylancl 662 . . . . 5  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  A  e.  _V )
3518, 34rabexd 4585 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  e.  _V )
36 ssrab2 3567 . . . . . 6  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
3718, 36eqsstri 3516 . . . . 5  |-  D  C_  A
3837a1i 11 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  C_  A
)
39 simprr 756 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  =/=  (/) )
40 wereu 4861 . . . 4  |-  ( ( w  We  A  /\  ( D  e.  _V  /\  D  C_  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
4129, 35, 38, 39, 40syl13anc 1229 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
42 riotacl 6253 . . 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 )
4341, 42syl 16 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e.  D
)
4428, 43eqeltrd 2529 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 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E!wreu 2793   {crab 2795   _Vcvv 3093    C_ wss 3458   (/)c0 3767   class class class wbr 4433    |-> cmpt 4491    Or wor 4785    We wwe 4823   Oncon0 4864   ran crn 4986    |` cres 4987   "cima 4988   Fun wfun 5568    Fn wfn 5569   ` cfv 5574   iota_crio 6237  recscrecs 7039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-recs 7040
This theorem is referenced by:  zorn2lem2  8875  zorn2lem3  8876  zorn2lem4  8877  zorn2lem5  8878
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