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Theorem noinfepOLD 7377
Description: Using the Axiom of Regularity in the form zfregfr 7332, show that there are no infinite descending 
e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
noinfepOLD  |-  ( F  Fn  om  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) )
Distinct variable group:    x, F

Proof of Theorem noinfepOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fndm 5359 . . . . 5  |-  ( F  Fn  om  ->  dom  F  =  om )
2 omex 7360 . . . . 5  |-  om  e.  _V
31, 2syl6eqel 2384 . . . 4  |-  ( F  Fn  om  ->  dom  F  e.  _V )
4 fnfun 5357 . . . 4  |-  ( F  Fn  om  ->  Fun  F )
5 funrnex 5763 . . . 4  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
63, 4, 5sylc 56 . . 3  |-  ( F  Fn  om  ->  ran  F  e.  _V )
7 peano1 4691 . . . . . . 7  |-  (/)  e.  om
8 eleq2 2357 . . . . . . 7  |-  ( dom 
F  =  om  ->  (
(/)  e.  dom  F  <->  (/)  e.  om ) )
97, 8mpbiri 224 . . . . . 6  |-  ( dom 
F  =  om  ->  (/)  e.  dom  F )
10 ne0i 3474 . . . . . 6  |-  ( (/)  e.  dom  F  ->  dom  F  =/=  (/) )
119, 10syl 15 . . . . 5  |-  ( dom 
F  =  om  ->  dom 
F  =/=  (/) )
12 dm0rn0 4911 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2491 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 188 . . . 4  |-  ( dom 
F  =  om  ->  ran 
F  =/=  (/) )
151, 14syl 15 . . 3  |-  ( F  Fn  om  ->  ran  F  =/=  (/) )
16 zfregfr 7332 . . . 4  |-  _E  Fr  ran  F
17 ssid 3210 . . . . 5  |-  ran  F  C_ 
ran  F
18 fri 4371 . . . . 5  |-  ( ( ( ran  F  e. 
_V  /\  _E  Fr  ran  F )  /\  ( ran  F  C_  ran  F  /\  ran  F  =/=  (/) ) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
1917, 18mpanr1 664 . . . 4  |-  ( ( ( ran  F  e. 
_V  /\  _E  Fr  ran  F )  /\  ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
2016, 19mpanl2 662 . . 3  |-  ( ( ran  F  e.  _V  /\ 
ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
216, 15, 20syl2anc 642 . 2  |-  ( F  Fn  om  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
22 fvelrnb 5586 . . . . . . 7  |-  ( F  Fn  om  ->  (
y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
2322adantr 451 . . . . . 6  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
24 peano2 4692 . . . . . . . . 9  |-  ( x  e.  om  ->  suc  x  e.  om )
25 fnfvelrn 5678 . . . . . . . . . . . 12  |-  ( ( F  Fn  om  /\  suc  x  e.  om )  ->  ( F `  suc  x )  e.  ran  F )
2625adantlr 695 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  ( F `  suc  x )  e.  ran  F )
27 simplr 731 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  A. z  e.  ran  F  -.  z  _E  y )
2826, 27jca 518 . . . . . . . . . 10  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  (
( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y ) )
29 epel 4324 . . . . . . . . . . . . . 14  |-  ( z  _E  y  <->  z  e.  y )
30 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3129, 30syl5bb 248 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3231notbid 285 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
3332rspcva 2895 . . . . . . . . . . 11  |-  ( ( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  y )
34 eleq2 2357 . . . . . . . . . . . 12  |-  ( ( F `  x )  =  y  ->  (
( F `  suc  x )  e.  ( F `  x )  <-> 
( F `  suc  x )  e.  y ) )
3534notbid 285 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  ->  ( -.  ( F `  suc  x )  e.  ( F `  x )  <->  -.  ( F `  suc  x )  e.  y ) )
3633, 35syl5ibr 212 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  ( F `
 x ) ) )
3728, 36syl5 28 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( ( F  Fn  om 
/\  A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  -.  ( F `  suc  x
)  e.  ( F `
 x ) ) )
3824, 37sylan2i 636 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( ( F  Fn  om 
/\  A. z  e.  ran  F  -.  z  _E  y
)  /\  x  e.  om )  ->  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
3938com12 27 . . . . . . 7  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  x  e.  om )  ->  ( ( F `  x )  =  y  ->  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
4039reximdva 2668 . . . . . 6  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( E. x  e.  om  ( F `  x )  =  y  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) )
4123, 40sylbid 206 . . . . 5  |-  ( ( F  Fn  om  /\  A. z  e.  ran  F  -.  z  _E  y
)  ->  ( y  e.  ran  F  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) )
4241ex 423 . . . 4  |-  ( F  Fn  om  ->  ( A. z  e.  ran  F  -.  z  _E  y  ->  ( y  e.  ran  F  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) ) ) )
4342com23 72 . . 3  |-  ( F  Fn  om  ->  (
y  e.  ran  F  ->  ( A. z  e. 
ran  F  -.  z  _E  y  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) ) ) )
4443rexlimdv 2679 . 2  |-  ( F  Fn  om  ->  ( E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) ) )
4521, 44mpd 14 1  |-  ( F  Fn  om  ->  E. x  e.  om  -.  ( F `
 suc  x )  e.  ( F `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039    _E cep 4319    Fr wfr 4365   suc csuc 4410   omcom 4672   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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