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Theorem zorn2lem1 8865
Description: Lemma for zorn2 8875. (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 7057 . . . 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 7056 . . . . . 6  |-  F  Fn  On
5 fnfun 5669 . . . . . 6  |-  ( F  Fn  On  ->  Fun  F )
64, 5ax-mp 5 . . . . 5  |-  Fun  F
7 vex 3109 . . . . 5  |-  x  e. 
_V
8 resfunexg 6117 . . . . 5  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
96, 7, 8mp2an 672 . . . 4  |-  ( F  |`  x )  e.  _V
10 rneq 5219 . . . . . . . . . . . 12  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
11 df-ima 5005 . . . . . . . . . . . 12  |-  ( F
" x )  =  ran  ( F  |`  x )
1210, 11syl6eqr 2519 . . . . . . . . . . 11  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
1312eleq2d 2530 . . . . . . . . . 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 2892 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  ( A. g  e.  ran  f  g R z  <->  A. g  e.  ( F " x ) g R z ) )
1615rabbidv 3098 . . . . . . 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 2526 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  C  =  D )
2019eleq2d 2530 . . . . . . . 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 2892 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  ( A. u  e.  C  -.  u w v  <->  A. u  e.  D  -.  u w v ) )
2319, 22riotaeqbidv 6239 . . . . 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 2460 . . . . 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 6240 . . . . 5  |-  ( iota_ v  e.  D  A. u  e.  D  -.  u w v )  e. 
_V
2623, 24, 25fvmpt 5941 . . . 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 2517 . 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 3578 . . . . . 6  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
3118, 30eqsstri 3527 . . . . 5  |-  D  C_  A
32 weso 4863 . . . . . . 7  |-  ( w  We  A  ->  w  Or  A )
3332ad2antrl 727 . . . . . 6  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  Or  A
)
34 vex 3109 . . . . . 6  |-  w  e. 
_V
35 soex 6717 . . . . . 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 4586 . . . . 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 4868 . . . 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 1225 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
43 riotacl 6251 . . 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 2548 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 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E!wreu 2809   {crab 2811   _Vcvv 3106    C_ wss 3469   (/)c0 3778   class class class wbr 4440    |-> cmpt 4498    Or wor 4792    We wwe 4830   Oncon0 4871   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573    Fn wfn 5574   ` cfv 5579   iota_crio 6235  recscrecs 7031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-recs 7032
This theorem is referenced by:  zorn2lem2  8866  zorn2lem3  8867  zorn2lem4  8868  zorn2lem5  8869
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