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Theorem zorn2lem4 8381
Description: Lemma for zorn2 8388. (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 pm3.24 854 . 2  |-  -.  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V )
2 df-ne 2603 . . . . 5  |-  ( D  =/=  (/)  <->  -.  D  =  (/) )
32ralbii 2731 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x  e.  On  -.  D  =  (/) )
4 df-ral 2712 . . . 4  |-  ( A. x  e.  On  D  =/=  (/)  <->  A. x ( x  e.  On  ->  D  =/=  (/) ) )
5 ralnex 2717 . . . 4  |-  ( A. x  e.  On  -.  D  =  (/)  <->  -.  E. x  e.  On  D  =  (/) )
63, 4, 53bitr3i 268 . . 3  |-  ( A. x ( x  e.  On  ->  D  =/=  (/) )  <->  -.  E. x  e.  On  D  =  (/) )
7 weso 4575 . . . . . . . . 9  |-  ( w  We  A  ->  w  Or  A )
87adantr 453 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  w  Or  A )
9 vex 2961 . . . . . . . 8  |-  w  e. 
_V
10 soex 5321 . . . . . . . 8  |-  ( ( w  Or  A  /\  w  e.  _V )  ->  A  e.  _V )
118, 9, 10sylancl 645 . . . . . . 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 6660 . . . . . . . . . 10  |-  F  Fn  On
14 fvelrnb 5776 . . . . . . . . . 10  |-  ( F  Fn  On  ->  (
y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y ) )
1513, 14ax-mp 8 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  e.  On  ( F `  x )  =  y )
16 nfv 1630 . . . . . . . . . . 11  |-  F/ x  w  We  A
17 nfa1 1807 . . . . . . . . . . 11  |-  F/ x A. x ( x  e.  On  ->  D  =/=  (/) )
1816, 17nfan 1847 . . . . . . . . . 10  |-  F/ x
( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )
19 nfv 1630 . . . . . . . . . 10  |-  F/ x  y  e.  A
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
21 ssrab2 3430 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
2220, 21eqsstri 3380 . . . . . . . . . . . . . . . . 17  |-  D  C_  A
23 zorn2lem.4 . . . . . . . . . . . . . . . . . 18  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
2412, 23, 20zorn2lem1 8378 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
2522, 24sseldi 3348 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
26 eleq1 2498 . . . . . . . . . . . . . . . 16  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2725, 26syl5ibcom 213 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( ( F `
 x )  =  y  ->  y  e.  A ) )
2827exp32 590 . . . . . . . . . . . . . 14  |-  ( x  e.  On  ->  (
w  We  A  -> 
( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
2928com12 30 . . . . . . . . . . . . 13  |-  ( w  We  A  ->  (
x  e.  On  ->  ( D  =/=  (/)  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) ) ) )
3029a2d 25 . . . . . . . . . . . 12  |-  ( w  We  A  ->  (
( x  e.  On  ->  D  =/=  (/) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) ) )
3130spsd 1772 . . . . . . . . . . 11  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A ) ) ) )
3231imp 420 . . . . . . . . . 10  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
x  e.  On  ->  ( ( F `  x
)  =  y  -> 
y  e.  A ) ) )
3318, 19, 32rexlimd 2829 . . . . . . . . 9  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ( E. x  e.  On  ( F `  x )  =  y  ->  y  e.  A ) )
3415, 33syl5bi 210 . . . . . . . 8  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  (
y  e.  ran  F  ->  y  e.  A ) )
3534ssrdv 3356 . . . . . . 7  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F 
C_  A )
3611, 35ssexd 4352 . . . . . 6  |-  ( ( w  We  A  /\  A. x ( x  e.  On  ->  D  =/=  (/) ) )  ->  ran  F  e.  _V )
3736ex 425 . . . . 5  |-  ( w  We  A  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e. 
_V ) )
3837adantl 454 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ran  F  e.  _V ) )
3912, 23, 20zorn2lem3 8380 . . . . . . . . . . . . . 14  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
4039exp45 599 . . . . . . . . . . . . 13  |-  ( R  Po  A  ->  (
x  e.  On  ->  ( w  We  A  -> 
( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4140com23 75 . . . . . . . . . . . 12  |-  ( R  Po  A  ->  (
w  We  A  -> 
( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) ) )
4241imp 420 . . . . . . . . . . 11  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( x  e.  On  ->  ( D  =/=  (/)  ->  (
y  e.  x  ->  -.  ( F `  x
)  =  ( F `
 y ) ) ) ) )
4342a2d 25 . . . . . . . . . 10  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( x  e.  On  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) ) ) )
4443imp4a 574 . . . . . . . . 9  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  ( (
x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4544alrimdv 1644 . . . . . . . 8  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( ( x  e.  On  ->  D  =/=  (/) )  ->  A. y
( ( x  e.  On  /\  y  e.  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) ) )
4645alimdv 1632 . . . . . . 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 2744 . . . . . . 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 220 . . . . . 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 3369 . . . . . . . 8  |-  On  C_  On
5013tz7.48lem 6700 . . . . . . . 8  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
5149, 50mpan 653 . . . . . . 7  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' ( F  |`  On ) )
52 fnrel 5545 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  Rel  F )
5313, 52ax-mp 8 . . . . . . . . . 10  |-  Rel  F
54 fndm 5546 . . . . . . . . . . . 12  |-  ( F  Fn  On  ->  dom  F  =  On )
5513, 54ax-mp 8 . . . . . . . . . . 11  |-  dom  F  =  On
5655eqimssi 3404 . . . . . . . . . 10  |-  dom  F  C_  On
57 relssres 5185 . . . . . . . . . 10  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
5853, 56, 57mp2an 655 . . . . . . . . 9  |-  ( F  |`  On )  =  F
5958cnveqi 5049 . . . . . . . 8  |-  `' ( F  |`  On )  =  `' F
6059funeqi 5476 . . . . . . 7  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
6151, 60sylib 190 . . . . . 6  |-  ( A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y )  ->  Fun  `' F )
6248, 61syl6 32 . . . . 5  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  Fun  `' F ) )
63 onprc 4767 . . . . . 6  |-  -.  On  e.  _V
64 funrnex 5969 . . . . . . . 8  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
6564com12 30 . . . . . . 7  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
66 df-rn 4891 . . . . . . . 8  |-  ran  F  =  dom  `' F
6766eleq1i 2501 . . . . . . 7  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
68 dfdm4 5065 . . . . . . . . 9  |-  dom  F  =  ran  `' F
6955, 68eqtr3i 2460 . . . . . . . 8  |-  On  =  ran  `' F
7069eleq1i 2501 . . . . . . 7  |-  ( On  e.  _V  <->  ran  `' F  e.  _V )
7165, 67, 703imtr4g 263 . . . . . 6  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  On  e.  _V ) )
7263, 71mtoi 172 . . . . 5  |-  ( Fun  `' F  ->  -.  ran  F  e.  _V )
7362, 72syl6 32 . . . 4  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  -.  ran  F  e.  _V )
)
7438, 73jcad 521 . . 3  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( A. x ( x  e.  On  ->  D  =/=  (/) )  ->  ( ran  F  e.  _V  /\  -.  ran  F  e.  _V ) ) )
756, 74syl5bir 211 . 2  |-  ( ( R  Po  A  /\  w  We  A )  ->  ( -.  E. x  e.  On  D  =  (/)  ->  ( ran  F  e. 
_V  /\  -.  ran  F  e.  _V ) ) )
761, 75mt3i 121 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 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   class class class wbr 4214    e. cmpt 4268    Po wpo 4503    Or wor 4504    We wwe 4542   Oncon0 4583   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Rel wrel 4885   Fun wfun 5450    Fn wfn 5451   ` cfv 5456   iota_crio 6544  recscrecs 6634
This theorem is referenced by:  zorn2lem7  8384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-riota 6551  df-recs 6635
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