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Theorem zorn2lem1 8376
Description: Lemma for zorn2 8386. (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 6659 . . . 4  |-  ( x  e.  On  ->  ( F `  x )  =  ( ( f  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u w v ) ) `
 ( F  |`  x ) ) )
32adantr 452 . . 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 6658 . . . . . 6  |-  F  Fn  On
5 fnfun 5542 . . . . . 6  |-  ( F  Fn  On  ->  Fun  F )
64, 5ax-mp 8 . . . . 5  |-  Fun  F
7 vex 2959 . . . . 5  |-  x  e. 
_V
8 resfunexg 5957 . . . . 5  |-  ( ( Fun  F  /\  x  e.  _V )  ->  ( F  |`  x )  e. 
_V )
96, 7, 8mp2an 654 . . . 4  |-  ( F  |`  x )  e.  _V
10 rneq 5095 . . . . . . . . . . . 12  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ran  ( F  |`  x ) )
11 df-ima 4891 . . . . . . . . . . . 12  |-  ( F
" x )  =  ran  ( F  |`  x )
1210, 11syl6eqr 2486 . . . . . . . . . . 11  |-  ( f  =  ( F  |`  x )  ->  ran  f  =  ( F " x ) )
1312eleq2d 2503 . . . . . . . . . 10  |-  ( f  =  ( F  |`  x )  ->  (
g  e.  ran  f  <->  g  e.  ( F "
x ) ) )
1413imbi1d 309 . . . . . . . . 9  |-  ( f  =  ( F  |`  x )  ->  (
( g  e.  ran  f  ->  g R z )  <->  ( g  e.  ( F " x
)  ->  g R
z ) ) )
1514ralbidv2 2727 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  ( A. g  e.  ran  f  g R z  <->  A. g  e.  ( F " x ) g R z ) )
1615rabbidv 2948 . . . . . . 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 2493 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  C  =  D )
2019eleq2d 2503 . . . . . . . 8  |-  ( f  =  ( F  |`  x )  ->  (
u  e.  C  <->  u  e.  D ) )
2120imbi1d 309 . . . . . . 7  |-  ( f  =  ( F  |`  x )  ->  (
( u  e.  C  ->  -.  u w v )  <->  ( u  e.  D  ->  -.  u w v ) ) )
2221ralbidv2 2727 . . . . . 6  |-  ( f  =  ( F  |`  x )  ->  ( A. u  e.  C  -.  u w v  <->  A. u  e.  D  -.  u w v ) )
2319, 22riotaeqbidv 6552 . . . . 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 2436 . . . . 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 6553 . . . . 5  |-  ( iota_ v  e.  D A. u  e.  D  -.  u w v )  e. 
_V
2623, 24, 25fvmpt 5806 . . . 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 2484 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  =  (
iota_ v  e.  D A. u  e.  D  -.  u w v ) )
29 simprl 733 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  We  A
)
30 ssrab2 3428 . . . . . 6  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
3118, 30eqsstri 3378 . . . . 5  |-  D  C_  A
32 weso 4573 . . . . . . 7  |-  ( w  We  A  ->  w  Or  A )
3332ad2antrl 709 . . . . . 6  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  w  Or  A
)
34 vex 2959 . . . . . 6  |-  w  e. 
_V
35 soex 5319 . . . . . 6  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
3633, 34, 35sylancl 644 . . . . 5  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  A  e.  _V )
37 ssexg 4349 . . . . 5  |-  ( ( D  C_  A  /\  A  e.  _V )  ->  D  e.  _V )
3831, 36, 37sylancr 645 . . . 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 734 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  D  =/=  (/) )
41 wereu 4578 . . . 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 1186 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  E! v  e.  D  A. u  e.  D  -.  u w v )
43 riotacl 6564 . . 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 2510 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E!wreu 2707   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   class class class wbr 4212    e. cmpt 4266    Or wor 4502    We wwe 4540   Oncon0 4581   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454   iota_crio 6542  recscrecs 6632
This theorem is referenced by:  zorn2lem2  8377  zorn2lem3  8378  zorn2lem4  8379  zorn2lem5  8380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-recs 6633
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