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

Theorem zorn2lem4 8926
Description: Lemma for zorn2 8933. (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 pm3.24 892 . 2  |-  -.  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V )
2 df-ne 2623 . . . . 5  |-  ( D  =/=  (/)  <->  -.  D  =  (/) )
32ralbii 2818 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x  e.  On  -.  D  =  (/) )
4 df-ral 2741 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x ( x  e.  On  ->  D  =/=  (/) ) )
5 ralnex 2833 . . . 4  |-  ( A. x  e.  On  -.  D  =  (/)  <->  -.  E. x  e.  On  D  =  (/) )
63, 4, 53bitr3i 279 . . 3  |-  ( A. x ( x  e.  On  ->  D  =/=  (/) )  <->  -.  E. x  e.  On  D  =  (/) )
7 weso 4824 . . . . . . . . 9  |-  ( w  We  A  ->  w  Or  A )
87adantr 467 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  w  Or  A )
9 vex 3047 . . . . . . . 8  |-  w  e. 
_V
10 soex 6733 . . . . . . . 8  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
118, 9, 10sylancl 667 . . . . . . 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 7112 . . . . . . . . . 10  |-  F  Fn  On
14 fvelrnb 5910 . . . . . . . . . 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 1760 . . . . . . . . . . 11  |-  F/ x  w  We  A
17 nfa1 1978 . . . . . . . . . . 11  |-  F/ x A. x ( x  e.  On  ->  D  =/=  (/) )
1816, 17nfan 2010 . . . . . . . . . 10  |-  F/ x
( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )
19 nfv 1760 . . . . . . . . . 10  |-  F/ x  y  e.  A
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
21 ssrab2 3513 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
2220, 21eqsstri 3461 . . . . . . . . . . . . . . . . 17  |-  D  C_  A
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
2412, 23, 20zorn2lem1 8923 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
2522, 24sseldi 3429 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
26 eleq1 2516 . . . . . . . . . . . . . . . 16  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2725, 26syl5ibcom 224 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( ( F `
 x )  =  y  ->  y  e.  A ) )
2827exp32 609 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  (
w  We  A  -> 
( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
2928com12 32 . . . . . . . . . . . . 13  |-  ( w  We  A  ->  (
x  e.  On  ->  ( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
3029a2d 29 . . . . . . . . . . . 12  |-  ( w  We  A  ->  (
( x  e.  On  ->  D  =/=  (/) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) ) )
3130spsd 1944 . . . . . . . . . . 11  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A ) ) ) )
3231imp 431 . . . . . . . . . 10  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
x  e.  On  ->  ( ( F `  x
)  =  y  -> 
y  e.  A ) ) )
3318, 19, 32rexlimd 2870 . . . . . . . . 9  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ( E. x  e.  On  ( F `  x )  =  y  ->  y  e.  A ) )
3415, 33syl5bi 221 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
y  e.  ran  F  ->  y  e.  A ) )
3534ssrdv 3437 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F 
C_  A )
3611, 35ssexd 4549 . . . . . 6  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F  e.  _V )
3736ex 436 . . . . 5  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e. 
_V ) )
3837adantl 468 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e.  _V ) )
3912, 23, 20zorn2lem3 8925 . . . . . . . . . . . . . 14  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
4039exp45 618 . . . . . . . . . . . . 13  |-  ( R  Po  A  ->  (
x  e.  On  ->  ( w  We  A  -> 
( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4140com23 81 . . . . . . . . . . . 12  |-  ( R  Po  A  ->  (
w  We  A  -> 
( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4241imp 431 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) )
4342a2d 29 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) ) ) )
4443imp4a 593 . . . . . . . . 9  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( (
x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4544alrimdv 1774 . . . . . . . 8  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  A. y
( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4645alimdv 1762 . . . . . . 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 2765 . . . . . . 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 231 . . . . . 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 3450 . . . . . . . 8  |-  On  C_  On
5013tz7.48lem 7155 . . . . . . . 8  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
5149, 50mpan 675 . . . . . . 7  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' ( F  |`  On ) )
52 fnrel 5672 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Rel  F )
5313, 52ax-mp 5 . . . . . . . . . 10  |-  Rel  F
54 fndm 5673 . . . . . . . . . . . 12  |-  ( F  Fn  On  ->  dom  F  =  On )
5513, 54ax-mp 5 . . . . . . . . . . 11  |-  dom  F  =  On
5655eqimssi 3485 . . . . . . . . . 10  |-  dom  F  C_  On
57 relssres 5141 . . . . . . . . . 10  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
5853, 56, 57mp2an 677 . . . . . . . . 9  |-  ( F  |`  On )  =  F
5958cnveqi 5008 . . . . . . . 8  |-  `' ( F  |`  On )  =  `' F
6059funeqi 5601 . . . . . . 7  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
6151, 60sylib 200 . . . . . 6  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' F )
6248, 61syl6 34 . . . . 5  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  Fun  `' F ) )
63 onprc 6608 . . . . . 6  |-  -.  On  e.  _V
64 funrnex 6757 . . . . . . . 8  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
6564com12 32 . . . . . . 7  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
66 df-rn 4844 . . . . . . . 8  |-  ran  F  =  dom  `' F
6766eleq1i 2519 . . . . . . 7  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
68 dfdm4 5026 . . . . . . . . 9  |-  dom  F  =  ran  `' F
6955, 68eqtr3i 2474 . . . . . . . 8  |-  On  =  ran  `' F
7069eleq1i 2519 . . . . . . 7  |-  ( On  e.  _V  <->  ran  `' F  e.  _V )
7165, 67, 703imtr4g 274 . . . . . 6  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  On  e.  _V ) )
7263, 71mtoi 182 . . . . 5  |-  ( Fun  `' F  ->  -.  ran  F  e.  _V )
7362, 72syl6 34 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  -.  ran  F  e.  _V )
)
7438, 73jcad 536 . . 3  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V ) ) )
756, 74syl5bir 222 . 2  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( -.  E. x  e.  On  D  =  (/)  ->  ( ran  F  e. 
_V  /\  -.  ran  F  e.  _V ) ) )
761, 75mt3i 130 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 188    /\ wa 371   A.wal 1441    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044    C_ wss 3403   (/)c0 3730   class class class wbr 4401    |-> cmpt 4460    Po wpo 4752    Or wor 4753    We wwe 4791   `'ccnv 4832   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836   Rel wrel 4838   Oncon0 5422   Fun wfun 5575    Fn wfn 5576   ` cfv 5581   iota_crio 6249  recscrecs 7086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-wrecs 7025  df-recs 7087
This theorem is referenced by:  zorn2lem7  8929
  Copyright terms: Public domain W3C validator