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Theorem zorn2lem4 8128
Description: Lemma for zorn2 8135. (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 pm3.24 852 . 2  |-  -.  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V )
2 df-ne 2450 . . . . 5  |-  ( D  =/=  (/)  <->  -.  D  =  (/) )
32ralbii 2569 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x  e.  On  -.  D  =  (/) )
4 df-ral 2550 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x ( x  e.  On  ->  D  =/=  (/) ) )
5 ralnex 2555 . . . 4  |-  ( A. x  e.  On  -.  D  =  (/)  <->  -.  E. x  e.  On  D  =  (/) )
63, 4, 53bitr3i 266 . . 3  |-  ( A. x ( x  e.  On  ->  D  =/=  (/) )  <->  -.  E. x  e.  On  D  =  (/) )
7 zorn2lem.3 . . . . . . . . . . 11  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )
87tfr1 6415 . . . . . . . . . 10  |-  F  Fn  On
9 fvelrnb 5572 . . . . . . . . . 10  |-  ( F  Fn  On  ->  (
y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y ) )
108, 9ax-mp 8 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y )
11 nfv 1607 . . . . . . . . . . 11  |-  F/ x  w  We  A
12 nfa1 1758 . . . . . . . . . . 11  |-  F/ x A. x ( x  e.  On  ->  D  =/=  (/) )
1311, 12nfan 1773 . . . . . . . . . 10  |-  F/ x
( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )
14 nfv 1607 . . . . . . . . . 10  |-  F/ x  y  e.  A
15 zorn2lem.5 . . . . . . . . . . . . . . . . . 18  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
16 ssrab2 3260 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
1715, 16eqsstri 3210 . . . . . . . . . . . . . . . . 17  |-  D  C_  A
18 zorn2lem.4 . . . . . . . . . . . . . . . . . 18  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
197, 18, 15zorn2lem1 8125 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
2017, 19sseldi 3180 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
21 eleq1 2345 . . . . . . . . . . . . . . . 16  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2220, 21syl5ibcom 211 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( ( F `
 x )  =  y  ->  y  e.  A ) )
2322exp32 588 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  (
w  We  A  -> 
( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
2423com12 27 . . . . . . . . . . . . 13  |-  ( w  We  A  ->  (
x  e.  On  ->  ( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
2524a2d 23 . . . . . . . . . . . 12  |-  ( w  We  A  ->  (
( x  e.  On  ->  D  =/=  (/) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) ) )
2625spsd 1742 . . . . . . . . . . 11  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A ) ) ) )
2726imp 418 . . . . . . . . . 10  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
x  e.  On  ->  ( ( F `  x
)  =  y  -> 
y  e.  A ) ) )
2813, 14, 27rexlimd 2666 . . . . . . . . 9  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ( E. x  e.  On  ( F `  x )  =  y  ->  y  e.  A ) )
2910, 28syl5bi 208 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
y  e.  ran  F  ->  y  e.  A ) )
3029ssrdv 3187 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F 
C_  A )
31 weso 4386 . . . . . . . . 9  |-  ( w  We  A  ->  w  Or  A )
3231adantr 451 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  w  Or  A )
33 vex 2793 . . . . . . . 8  |-  w  e. 
_V
34 soex 5124 . . . . . . . 8  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
3532, 33, 34sylancl 643 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  A  e.  _V )
36 ssexg 4162 . . . . . . 7  |-  ( ( ran  F  C_  A  /\  A  e.  _V )  ->  ran  F  e.  _V )
3730, 35, 36syl2anc 642 . . . . . 6  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F  e.  _V )
3837ex 423 . . . . 5  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e. 
_V ) )
3938adantl 452 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e.  _V ) )
407, 18, 15zorn2lem3 8127 . . . . . . . . . . . . . 14  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
4140exp45 597 . . . . . . . . . . . . 13  |-  ( R  Po  A  ->  (
x  e.  On  ->  ( w  We  A  -> 
( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4241com23 72 . . . . . . . . . . . 12  |-  ( R  Po  A  ->  (
w  We  A  -> 
( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4342imp 418 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) )
4443a2d 23 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) ) ) )
4544imp4a 572 . . . . . . . . 9  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( (
x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4645alrimdv 1621 . . . . . . . 8  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  A. y
( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4746alimdv 1609 . . . . . . 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 ) ) ) )
48 r2al 2582 . . . . . . 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 ) ) )
4947, 48syl6ibr 218 . . . . . 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
) ) )
50 ssid 3199 . . . . . . . 8  |-  On  C_  On
518tz7.48lem 6455 . . . . . . . 8  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
5250, 51mpan 651 . . . . . . 7  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' ( F  |`  On ) )
53 fnrel 5344 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Rel  F )
548, 53ax-mp 8 . . . . . . . . . 10  |-  Rel  F
55 fndm 5345 . . . . . . . . . . . 12  |-  ( F  Fn  On  ->  dom  F  =  On )
568, 55ax-mp 8 . . . . . . . . . . 11  |-  dom  F  =  On
5756eqimssi 3234 . . . . . . . . . 10  |-  dom  F  C_  On
58 relssres 4994 . . . . . . . . . 10  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
5954, 57, 58mp2an 653 . . . . . . . . 9  |-  ( F  |`  On )  =  F
6059cnveqi 4858 . . . . . . . 8  |-  `' ( F  |`  On )  =  `' F
6160funeqi 5277 . . . . . . 7  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
6252, 61sylib 188 . . . . . 6  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' F )
6349, 62syl6 29 . . . . 5  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  Fun  `' F ) )
64 onprc 4578 . . . . . 6  |-  -.  On  e.  _V
65 df-rn 4702 . . . . . . . . 9  |-  ran  F  =  dom  `' F
6665eleq1i 2348 . . . . . . . 8  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
67 funrnex 5749 . . . . . . . . 9  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
6867com12 27 . . . . . . . 8  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
6966, 68syl5bi 208 . . . . . . 7  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  ran  `' F  e.  _V ) )
70 dfdm4 4874 . . . . . . . . 9  |-  dom  F  =  ran  `' F
7156, 70eqtr3i 2307 . . . . . . . 8  |-  On  =  ran  `' F
7271eleq1i 2348 . . . . . . 7  |-  ( On  e.  _V  <->  ran  `' F  e.  _V )
7369, 72syl6ibr 218 . . . . . 6  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  On  e.  _V ) )
7464, 73mtoi 169 . . . . 5  |-  ( Fun  `' F  ->  -.  ran  F  e.  _V )
7563, 74syl6 29 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  -.  ran  F  e.  _V )
)
7639, 75jcad 519 . . 3  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V ) ) )
776, 76syl5bir 209 . 2  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( -.  E. x  e.  On  D  =  (/)  ->  ( ran  F  e. 
_V  /\  -.  ran  F  e.  _V ) ) )
781, 77mt3i 118 1  |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  D  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    C_ wss 3154   (/)c0 3457   class class class wbr 4025    e. cmpt 4079    Po wpo 4314    Or wor 4315    We wwe 4353   Oncon0 4394   `'ccnv 4690   dom cdm 4691   ran crn 4692    |` cres 4693   "cima 4694   Rel wrel 4696   Fun wfun 5251    Fn wfn 5252   ` cfv 5257   iota_crio 6299  recscrecs 6389
This theorem is referenced by:  zorn2lem7  8131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-riota 6306  df-recs 6390
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