Theorem noinfep 4786

Description: Using the Axiom of Regularity in the form zfregfr 4746, show that there are no infinite descending e. -chains. Proposition 7.34 of [TakeutiZaring] p. 44.

Assertion

Ref	Expression
noinfep	|- (F Fn om -> E.x e. om -. (F` suc x) e. (F` x))

Distinct variable group:   x,F

Proof of Theorem noinfep

Step	Hyp	Ref	Expression
1		zfregfr 4746	. . . 4 |- E Fr ran F
2		ssid 2132	. . . . 5 |- ran F (_ ran F
3		fri 2948	. . . . 5 |- (((ran F e. V /\ E Fr ran F) /\ (ran F (_ ran F /\ ran F =/= (/))) -> E.y e. ran FA.z e. ran F -. zEy)
4	2, 3	mpanr1 713	. . . 4 |- (((ran F e. V /\ E Fr ran F) /\ ran F =/= (/)) -> E.y e. ran FA.z e. ran F -. zEy)
5	1, 4	mpanl2 711	. . 3 |- ((ran F e. V /\ ran F =/= (/)) -> E.y e. ran FA.z e. ran F -. zEy)
6		funrnex 3720	. . . 4 |- (dom F e. V -> (Fun F -> ran F e. V))
7		fndm 3693	. . . . 5 |- (F Fn om -> dom F = om)
8		omex 4772	. . . . 5 |- om e. V
9	7, 8	syl6eqel 1599	. . . 4 |- (F Fn om -> dom F e. V)
10		fnfun 3691	. . . 4 |- (F Fn om -> Fun F)
11	6, 9, 10	sylc 68	. . 3 |- (F Fn om -> ran F e. V)
12		peano1 3237	. . . . . . 7 |- (/) e. om
13		eleq2 1578	. . . . . . 7 |- (dom F = om -> ((/) e. dom F <-> (/) e. om))
14	12, 13	mpbiri 192	. . . . . 6 |- (dom F = om -> (/) e. dom F)
15		ne0i 2338	. . . . . 6 |- ((/) e. dom F -> dom F =/= (/))
16	14, 15	syl 10	. . . . 5 |- (dom F = om -> dom F =/= (/))
17		dm0rn0 3417	. . . . . 6 |- (dom F = (/) <-> ran F = (/))
18	17	necon3bii 1641	. . . . 5 |- (dom F =/= (/) <-> ran F =/= (/))
19	16, 18	sylib 196	. . . 4 |- (dom F = om -> ran F =/= (/))
20	7, 19	syl 10	. . 3 |- (F Fn om -> ran F =/= (/))
21	5, 11, 20	sylanc 473	. 2 |- (F Fn om -> E.y e. ran FA.z e. ran F -. zEy)
22		fvelrnb 3871	. . . . . . 7 |- (F Fn om -> (y e. ran F <-> E.x e. om (F` x) = y))
23	22	adantr 389	. . . . . 6 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (y e. ran F <-> E.x e. om (F` x) = y))
24		eleq2 1578	. . . . . . . . . . . 12 |- ((F` x) = y -> ((F` suc x) e. (F` x) <-> (F` suc x) e. y))
25	24	notbid 614	. . . . . . . . . . 11 |- ((F` x) = y -> (-. (F` suc x) e. (F` x) <-> -. (F` suc x) e. y))
26		eleq1 1577	. . . . . . . . . . . . . 14 |- (z = (F` suc x) -> (z e. y <-> (F` suc x) e. y))
27		epel 2912	. . . . . . . . . . . . . 14 |- (zEy <-> z e. y)
28	26, 27	syl5bb 535	. . . . . . . . . . . . 13 |- (z = (F` suc x) -> (zEy <-> (F` suc x) e. y))
29	28	notbid 614	. . . . . . . . . . . 12 |- (z = (F` suc x) -> (-. zEy <-> -. (F` suc x) e. y))
30	29	rcla4va 1921	. . . . . . . . . . 11 |- (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. y)
31	25, 30	syl5bir 208	. . . . . . . . . 10 |- ((F` x) = y -> (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. (F` x)))
32		fnfvelrn 3927	. . . . . . . . . . . 12 |- ((F Fn om /\ suc x e. om) -> (F` suc x) e. ran F)
33	32	adantlr 393	. . . . . . . . . . 11 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> (F` suc x) e. ran F)
34		simplr 413	. . . . . . . . . . 11 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> A.z e. ran F -. zEy)
35	33, 34	jca 286	. . . . . . . . . 10 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> ((F` suc x) e. ran F /\ A.z e. ran F -. zEy))
36	31, 35	syl5 21	. . . . . . . . 9 |- ((F` x) = y -> (((F Fn om /\ A.z e. ran F -. zEy) /\ suc x e. om) -> -. (F` suc x) e. (F` x)))
37		peano2 3238	. . . . . . . . 9 |- (x e. om -> suc x e. om)
38	36, 37	sylan2i 467	. . . . . . . 8 |- ((F` x) = y -> (((F Fn om /\ A.z e. ran F -. zEy) /\ x e. om) -> -. (F` suc x) e. (F` x)))
39	38	com12 11	. . . . . . 7 |- (((F Fn om /\ A.z e. ran F -. zEy) /\ x e. om) -> ((F` x) = y -> -. (F` suc x) e. (F` x)))
40	39	r19.22dva 1785	. . . . . 6 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (E.x e. om (F` x) = y -> E.x e. om -. (F` suc x) e. (F` x)))
41	23, 40	sylbid 201	. . . . 5 |- ((F Fn om /\ A.z e. ran F -. zEy) -> (y e. ran F -> E.x e. om -. (F` suc x) e. (F` x)))
42	41	ex 371	. . . 4 |- (F Fn om -> (A.z e. ran F -. zEy -> (y e. ran F -> E.x e. om -. (F` suc x) e. (F` x))))
43	42	com23 32	. . 3 |- (F Fn om -> (y e. ran F -> (A.z e. ran F -. zEy -> E.x e. om -. (F` suc x) e. (F` x))))
44	43	r19.23adv 1792	. 2 |- (F Fn om -> (E.y e. ran FA.z e. ran F -. zEy -> E.x e. om -. (F` suc x) e. (F` x)))
45	21, 44	mpd 26	1 |- (F Fn om -> E.x e. om -. (F` suc x) e. (F` x))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628  A.wral 1691  E.wrex 1692  Vcvv 1857   (_ wss 2099  (/)c0 2332   class class class wbr 2692  Ecep 2908   Fr wfr 2945  suc csuc 2977  omcom 3218  dom cdm 3251  ran crn 3252  Fun wfun 3257   Fn wfn 3258  ` cfv 3263

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-reg 4736  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279

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