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Theorem zorn2lem1 8139
Description: Lemma for zorn2 8149. (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 6430 . . . 4  |-  ( x  e.  On  ->  ( F `  x )  =  ( ( f  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u w v ) ) `
 ( F  |`  x ) ) )
32adantr 451 . . 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 6429 . . . . . 6  |-  F  Fn  On
5 fnfun 5357 . . . . . 6  |-  ( F  Fn  On  ->  Fun  F )
64, 5ax-mp 8 . . . . 5  |-  Fun  F
7 vex 2804 . . . . 5  |-  x  e. 
_V
8 resfunexg 5753 . . . . 5  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
96, 7, 8mp2an 653 . . . 4  |-  ( F  |`  x )  e.  _V
10 rneq 4920 . . . . . . . . . . . 12  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
11 df-ima 4718 . . . . . . . . . . . 12  |-  ( F
" x )  =  ran  ( F  |`  x )
1210, 11syl6eqr 2346 . . . . . . . . . . 11  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
1312eleq2d 2363 . . . . . . . . . 10  |-  ( f  =  ( F  |`  x )  ->  (
g  e.  ran  f  <->  g  e.  ( F "
x ) ) )
1413imbi1d 308 . . . . . . . . 9  |-  ( f  =  ( F  |`  x )  ->  (
( g  e.  ran  f  ->  g R z )  <->  ( g  e.  ( F " x
)  ->  g R
z ) ) )
1514ralbidv2 2578 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  ( A. g  e.  ran  f  g R z  <->  A. g  e.  ( F " x ) g R z ) )
1615rabbidv 2793 . . . . . . 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 2353 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  C  =  D )
2019eleq2d 2363 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  (
u  e.  C  <->  u  e.  D ) )
2120imbi1d 308 . . . . . . 7  |-  ( f  =  ( F  |`  x )  ->  (
( u  e.  C  ->  -.  u w v )  <->  ( u  e.  D  ->  -.  u w v ) ) )
2221ralbidv2 2578 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  ( A. u  e.  C  -.  u w v  <->  A. u  e.  D  -.  u w v ) )
2319, 22riotaeqbidv 6323 . . . . 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 2296 . . . . 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 6324 . . . . 5  |-  ( iota_ v  e.  D A. u  e.  D  -.  u w v )  e. 
_V
2623, 24, 25fvmpt 5618 . . . 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 8 . . 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 2344 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  =  (
iota_ v  e.  D A. u  e.  D  -.  u w v ) )
29 simprl 732 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  We  A
)
30 ssrab2 3271 . . . . . 6  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
3118, 30eqsstri 3221 . . . . 5  |-  D  C_  A
32 weso 4400 . . . . . . 7  |-  ( w  We  A  ->  w  Or  A )
3332ad2antrl 708 . . . . . 6  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  Or  A
)
34 vex 2804 . . . . . 6  |-  w  e. 
_V
35 soex 5138 . . . . . 6  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
3633, 34, 35sylancl 643 . . . . 5  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  A  e.  _V )
37 ssexg 4176 . . . . 5  |-  ( ( D  C_  A  /\  A  e.  _V )  ->  D  e.  _V )
3831, 36, 37sylancr 644 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  e.  _V )
3931a1i 10 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  C_  A
)
40 simprr 733 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  =/=  (/) )
41 wereu 4405 . . . 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 1184 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
43 riotacl 6335 . . 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 15 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( iota_ v  e.  D A. u  e.  D  -.  u w v )  e.  D
)
4528, 44eqeltrd 2370 1  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E!wreu 2558   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093    Or wor 4329    We wwe 4367   Oncon0 4408   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   iota_crio 6313  recscrecs 6403
This theorem is referenced by:  zorn2lem2  8140  zorn2lem3  8141  zorn2lem4  8142  zorn2lem5  8143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-recs 6404
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