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Theorem noinfepOLD 7871
Description: Using the Axiom of Regularity in the form zfregfr 7823, 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 5515 . . . . 5  |-  ( F  Fn  om  ->  dom  F  =  om )
2 omex 7854 . . . . 5  |-  om  e.  _V
31, 2syl6eqel 2531 . . . 4  |-  ( F  Fn  om  ->  dom  F  e.  _V )
4 fnfun 5513 . . . 4  |-  ( F  Fn  om  ->  Fun  F )
5 funrnex 6549 . . . 4  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
63, 4, 5sylc 60 . . 3  |-  ( F  Fn  om  ->  ran  F  e.  _V )
7 peano1 6500 . . . . . . 7  |-  (/)  e.  om
8 eleq2 2504 . . . . . . 7  |-  ( dom 
F  =  om  ->  (
(/)  e.  dom  F  <->  (/)  e.  om ) )
97, 8mpbiri 233 . . . . . 6  |-  ( dom 
F  =  om  ->  (/)  e.  dom  F )
10 ne0i 3648 . . . . . 6  |-  ( (/)  e.  dom  F  ->  dom  F  =/=  (/) )
119, 10syl 16 . . . . 5  |-  ( dom 
F  =  om  ->  dom 
F  =/=  (/) )
12 dm0rn0 5061 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1312necon3bii 2645 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
1411, 13sylib 196 . . . 4  |-  ( dom 
F  =  om  ->  ran 
F  =/=  (/) )
151, 14syl 16 . . 3  |-  ( F  Fn  om  ->  ran  F  =/=  (/) )
16 zfregfr 7823 . . . 4  |-  _E  Fr  ran  F
17 ssid 3380 . . . . 5  |-  ran  F  C_ 
ran  F
18 fri 4687 . . . . 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 683 . . . 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 681 . . 3  |-  ( ( ran  F  e.  _V  /\ 
ran  F  =/=  (/) )  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y
)
216, 15, 20syl2anc 661 . 2  |-  ( F  Fn  om  ->  E. y  e.  ran  F A. z  e.  ran  F  -.  z  _E  y )
22 fvelrnb 5744 . . . . . . 7  |-  ( F  Fn  om  ->  (
y  e.  ran  F  <->  E. x  e.  om  ( F `  x )  =  y ) )
2322adantr 465 . . . . . 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 6501 . . . . . . . . 9  |-  ( x  e.  om  ->  suc  x  e.  om )
25 fnfvelrn 5845 . . . . . . . . . . . 12  |-  ( ( F  Fn  om  /\  suc  x  e.  om )  ->  ( F `  suc  x )  e.  ran  F )
2625adantlr 714 . . . . . . . . . . 11  |-  ( ( ( F  Fn  om  /\ 
A. z  e.  ran  F  -.  z  _E  y
)  /\  suc  x  e. 
om )  ->  ( F `  suc  x )  e.  ran  F )
27 simplr 754 . . . . . . . . . . 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 532 . . . . . . . . . 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 4640 . . . . . . . . . . . . . 14  |-  ( z  _E  y  <->  z  e.  y )
30 eleq1 2503 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  suc  x )  ->  (
z  e.  y  <->  ( F `  suc  x )  e.  y ) )
3129, 30syl5bb 257 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  suc  x )  ->  (
z  _E  y  <->  ( F `  suc  x )  e.  y ) )
3231notbid 294 . . . . . . . . . . . 12  |-  ( z  =  ( F `  suc  x )  ->  ( -.  z  _E  y  <->  -.  ( F `  suc  x )  e.  y ) )
3332rspcva 3076 . . . . . . . . . . 11  |-  ( ( ( F `  suc  x )  e.  ran  F  /\  A. z  e. 
ran  F  -.  z  _E  y )  ->  -.  ( F `  suc  x
)  e.  y )
34 eleq2 2504 . . . . . . . . . . . 12  |-  ( ( F `  x )  =  y  ->  (
( F `  suc  x )  e.  ( F `  x )  <-> 
( F `  suc  x )  e.  y ) )
3534notbid 294 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  ->  ( -.  ( F `  suc  x )  e.  ( F `  x )  <->  -.  ( F `  suc  x )  e.  y ) )
3633, 35syl5ibr 221 . . . . . . . . . 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 32 . . . . . . . . 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 655 . . . . . . . 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 31 . . . . . . 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 2833 . . . . . 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 215 . . . . 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 434 . . . 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 78 . . 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 2845 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   class class class wbr 4297    _E cep 4635    Fr wfr 4681   suc csuc 4726   dom cdm 4845   ran crn 4846   Fun wfun 5417    Fn wfn 5418   ` cfv 5423   omcom 6481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482
This theorem is referenced by: (None)
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