MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn2lem4 Structured version   Unicode version

Theorem zorn2lem4 8870
Description: Lemma for zorn2 8877. (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
zorn2lem4  |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  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 zorn2lem4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pm3.24 880 . 2  |-  -.  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V )
2 df-ne 2651 . . . . 5  |-  ( D  =/=  (/)  <->  -.  D  =  (/) )
32ralbii 2885 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x  e.  On  -.  D  =  (/) )
4 df-ral 2809 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x ( x  e.  On  ->  D  =/=  (/) ) )
5 ralnex 2900 . . . 4  |-  ( A. x  e.  On  -.  D  =  (/)  <->  -.  E. x  e.  On  D  =  (/) )
63, 4, 53bitr3i 275 . . 3  |-  ( A. x ( x  e.  On  ->  D  =/=  (/) )  <->  -.  E. x  e.  On  D  =  (/) )
7 weso 4859 . . . . . . . . 9  |-  ( w  We  A  ->  w  Or  A )
87adantr 463 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  w  Or  A )
9 vex 3109 . . . . . . . 8  |-  w  e. 
_V
10 soex 6716 . . . . . . . 8  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
118, 9, 10sylancl 660 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  A  e.  _V )
12 zorn2lem.3 . . . . . . . . . . 11  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
1312tfr1 7058 . . . . . . . . . 10  |-  F  Fn  On
14 fvelrnb 5895 . . . . . . . . . 10  |-  ( F  Fn  On  ->  (
y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y ) )
1513, 14ax-mp 5 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y )
16 nfv 1712 . . . . . . . . . . 11  |-  F/ x  w  We  A
17 nfa1 1902 . . . . . . . . . . 11  |-  F/ x A. x ( x  e.  On  ->  D  =/=  (/) )
1816, 17nfan 1933 . . . . . . . . . 10  |-  F/ x
( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )
19 nfv 1712 . . . . . . . . . 10  |-  F/ x  y  e.  A
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
21 ssrab2 3571 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
2220, 21eqsstri 3519 . . . . . . . . . . . . . . . . 17  |-  D  C_  A
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
2412, 23, 20zorn2lem1 8867 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
2522, 24sseldi 3487 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
26 eleq1 2526 . . . . . . . . . . . . . . . 16  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2725, 26syl5ibcom 220 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( ( F `
 x )  =  y  ->  y  e.  A ) )
2827exp32 603 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  (
w  We  A  -> 
( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
2928com12 31 . . . . . . . . . . . . 13  |-  ( w  We  A  ->  (
x  e.  On  ->  ( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
3029a2d 26 . . . . . . . . . . . 12  |-  ( w  We  A  ->  (
( x  e.  On  ->  D  =/=  (/) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) ) )
3130spsd 1872 . . . . . . . . . . 11  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A ) ) ) )
3231imp 427 . . . . . . . . . 10  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
x  e.  On  ->  ( ( F `  x
)  =  y  -> 
y  e.  A ) ) )
3318, 19, 32rexlimd 2938 . . . . . . . . 9  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ( E. x  e.  On  ( F `  x )  =  y  ->  y  e.  A ) )
3415, 33syl5bi 217 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
y  e.  ran  F  ->  y  e.  A ) )
3534ssrdv 3495 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F 
C_  A )
3611, 35ssexd 4584 . . . . . 6  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F  e.  _V )
3736ex 432 . . . . 5  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e. 
_V ) )
3837adantl 464 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e.  _V ) )
3912, 23, 20zorn2lem3 8869 . . . . . . . . . . . . . 14  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
4039exp45 612 . . . . . . . . . . . . 13  |-  ( R  Po  A  ->  (
x  e.  On  ->  ( w  We  A  -> 
( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4140com23 78 . . . . . . . . . . . 12  |-  ( R  Po  A  ->  (
w  We  A  -> 
( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4241imp 427 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) )
4342a2d 26 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) ) ) )
4443imp4a 587 . . . . . . . . 9  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( (
x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4544alrimdv 1726 . . . . . . . 8  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  A. y
( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4645alimdv 1714 . . . . . . 7  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  A. x A. y ( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
47 r2al 2832 . . . . . . 7  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  <->  A. x A. y ( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
4846, 47syl6ibr 227 . . . . . 6  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  A. x  e.  On  A. y  e.  x  -.  ( F `
 x )  =  ( F `  y
) ) )
49 ssid 3508 . . . . . . . 8  |-  On  C_  On
5013tz7.48lem 7098 . . . . . . . 8  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
5149, 50mpan 668 . . . . . . 7  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' ( F  |`  On ) )
52 fnrel 5661 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Rel  F )
5313, 52ax-mp 5 . . . . . . . . . 10  |-  Rel  F
54 fndm 5662 . . . . . . . . . . . 12  |-  ( F  Fn  On  ->  dom  F  =  On )
5513, 54ax-mp 5 . . . . . . . . . . 11  |-  dom  F  =  On
5655eqimssi 3543 . . . . . . . . . 10  |-  dom  F  C_  On
57 relssres 5299 . . . . . . . . . 10  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
5853, 56, 57mp2an 670 . . . . . . . . 9  |-  ( F  |`  On )  =  F
5958cnveqi 5166 . . . . . . . 8  |-  `' ( F  |`  On )  =  `' F
6059funeqi 5590 . . . . . . 7  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
6151, 60sylib 196 . . . . . 6  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' F )
6248, 61syl6 33 . . . . 5  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  Fun  `' F ) )
63 onprc 6593 . . . . . 6  |-  -.  On  e.  _V
64 funrnex 6740 . . . . . . . 8  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
6564com12 31 . . . . . . 7  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
66 df-rn 4999 . . . . . . . 8  |-  ran  F  =  dom  `' F
6766eleq1i 2531 . . . . . . 7  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
68 dfdm4 5184 . . . . . . . . 9  |-  dom  F  =  ran  `' F
6955, 68eqtr3i 2485 . . . . . . . 8  |-  On  =  ran  `' F
7069eleq1i 2531 . . . . . . 7  |-  ( On  e.  _V  <->  ran  `' F  e.  _V )
7165, 67, 703imtr4g 270 . . . . . 6  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  On  e.  _V ) )
7263, 71mtoi 178 . . . . 5  |-  ( Fun  `' F  ->  -.  ran  F  e.  _V )
7362, 72syl6 33 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  -.  ran  F  e.  _V )
)
7438, 73jcad 531 . . 3  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V ) ) )
756, 74syl5bir 218 . 2  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( -.  E. x  e.  On  D  =  (/)  ->  ( ran  F  e. 
_V  /\  -.  ran  F  e.  _V ) ) )
761, 75mt3i 126 1  |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  D  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497    Po wpo 4787    Or wor 4788    We wwe 4826   Oncon0 4867   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Rel wrel 4993   Fun wfun 5564    Fn wfn 5565   ` cfv 5570   iota_crio 6231  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-recs 7034
This theorem is referenced by:  zorn2lem7  8873
  Copyright terms: Public domain W3C validator