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Theorem noinfepOLD 7607
Description: Using the Axiom of Regularity in the form zfregfr 7562, 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 5536 . . . . 5  |-  ( F  Fn  om  ->  dom  F  =  om )
2 omex 7590 . . . . 5  |-  om  e.  _V
31, 2syl6eqel 2523 . . . 4  |-  ( F  Fn  om  ->  dom  F  e.  _V )
4 fnfun 5534 . . . 4  |-  ( F  Fn  om  ->  Fun  F )
5 funrnex 5959 . . . 4  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
63, 4, 5sylc 58 . . 3  |-  ( F  Fn  om  ->  ran  F  e.  _V )
7 peano1 4856 . . . . . . 7  |-  (/)  e.  om
8 eleq2 2496 . . . . . . 7  |-  ( dom 
F  =  om  ->  (
(/)  e.  dom  F  <->  (/)  e.  om ) )
97, 8mpbiri 225 . . . . . 6  |-  ( dom 
F  =  om  ->  (/)  e.  dom  F )
10 ne0i 3626 . . . . . 6  |-  ( (/)  e.  dom  F  ->  dom  F  =/=  (/) )
119, 10syl 16 . . . . 5  |-  ( dom 
F  =  om  ->  dom 
F  =/=  (/) )
12 dm0rn0 5078 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2630 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 189 . . . 4  |-  ( dom 
F  =  om  ->  ran 
F  =/=  (/) )
151, 14syl 16 . . 3  |-  ( F  Fn  om  ->  ran  F  =/=  (/) )
16 zfregfr 7562 . . . 4  |-  _E  Fr  ran  F
17 ssid 3359 . . . . 5  |-  ran  F  C_ 
ran  F
18 fri 4536 . . . . 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 665 . . . 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 663 . . 3  |-  ( ( ran  F  e.  _V  /\ 
ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
216, 15, 20syl2anc 643 . 2  |-  ( F  Fn  om  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
22 fvelrnb 5766 . . . . . . 7  |-  ( F  Fn  om  ->  (
y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
2322adantr 452 . . . . . 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 4857 . . . . . . . . 9  |-  ( x  e.  om  ->  suc  x  e.  om )
25 fnfvelrn 5859 . . . . . . . . . . . 12  |-  ( ( F  Fn  om  /\  suc  x  e.  om )  ->  ( F `  suc  x )  e.  ran  F )
2625adantlr 696 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  ( F `  suc  x )  e.  ran  F )
27 simplr 732 . . . . . . . . . . 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 519 . . . . . . . . . 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 4489 . . . . . . . . . . . . . 14  |-  ( z  _E  y  <->  z  e.  y )
30 eleq1 2495 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3129, 30syl5bb 249 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3231notbid 286 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
3332rspcva 3042 . . . . . . . . . . 11  |-  ( ( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  y )
34 eleq2 2496 . . . . . . . . . . . 12  |-  ( ( F `  x )  =  y  ->  (
( F `  suc  x )  e.  ( F `  x )  <-> 
( F `  suc  x )  e.  y ) )
3534notbid 286 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  ->  ( -.  ( F `  suc  x )  e.  ( F `  x )  <->  -.  ( F `  suc  x )  e.  y ) )
3633, 35syl5ibr 213 . . . . . . . . . 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 30 . . . . . . . . 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 637 . . . . . . . 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 29 . . . . . . 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 2810 . . . . . 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 207 . . . . 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 424 . . . 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 74 . . 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 2821 . 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 15 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204    _E cep 4484    Fr wfr 4530   suc csuc 4575   omcom 4837   dom cdm 4870   ran crn 4871   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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