Theorem noinfep 4640

Description: Using the Axiom of Regularity in the form zfregfr 4601, 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 4601	. . . 4 |- E Fr ran F
2		ssid 2080	. . . . 5 |- ran F (_ ran F
3		fri 2918	. . . . 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 709	. . . 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 707	. . 3 |- ((ran F e. V /\ ran F =/= (/)) -> E.y e. ran FA.z e. ran F -. zEy)
6		funrnex 3613	. . . 4 |- (dom F e. V -> (Fun F -> ran F e. V))
7		fndm 3587	. . . . 5 |- (F Fn om -> dom F = om)
8		omex 4627	. . . . 5 |- om e. V
9	7, 8	syl6eqel 1556	. . . 4 |- (F Fn om -> dom F e. V)
10		fnfun 3585	. . . 4 |- (F Fn om -> Fun F)
11	6, 9, 10	sylc 68	. . 3 |- (F Fn om -> ran F e. V)
12		peano1 3149	. . . . . . 7 |- (/) e. om
13		eleq2 1535	. . . . . . 7 |- (dom F = om -> ((/) e. dom F <-> (/) e. om))
14	12, 13	mpbiri 194	. . . . . 6 |- (dom F = om -> (/) e. dom F)
15		ne0i 2286	. . . . . 6 |- ((/) e. dom F -> dom F =/= (/))
16	14, 15	syl 10	. . . . 5 |- (dom F = om -> dom F =/= (/))
17		dm0rn0 3330	. . . . . 6 |- (dom F = (/) <-> ran F = (/))
18	17	necon3bii 1598	. . . . 5 |- (dom F =/= (/) <-> ran F =/= (/))
19	16, 18	sylib 198	. . . 4 |- (dom F = om -> ran F =/= (/))
20	7, 19	syl 10	. . 3 |- (F Fn om -> ran F =/= (/))
21	5, 11, 20	sylanc 471	. 2 |- (F Fn om -> E.y e. ran FA.z e. ran F -. zEy)
22		fvelrnb 3760	. . . . . . 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 1535	. . . . . . . . . . . 12 |- ((F` x) = y -> ((F` suc x) e. (F` x) <-> (F` suc x) e. y))
25	24	negbid 611	. . . . . . . . . . 11 |- ((F` x) = y -> (-. (F` suc x) e. (F` x) <-> -. (F` suc x) e. y))
26		eleq1 1534	. . . . . . . . . . . . . 14 |- (z = (F` suc x) -> (z e. y <-> (F` suc x) e. y))
27		epel 2834	. . . . . . . . . . . . . 14 |- (zEy <-> z e. y)
28	26, 27	syl5bb 532	. . . . . . . . . . . . 13 |- (z = (F` suc x) -> (zEy <-> (F` suc x) e. y))
29	28	negbid 611	. . . . . . . . . . . 12 |- (z = (F` suc x) -> (-. zEy <-> -. (F` suc x) e. y))
30	29	rcla4va 1875	. . . . . . . . . . 11 |- (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. y)
31	25, 30	syl5bir 210	. . . . . . . . . 10 |- ((F` x) = y -> (((F` suc x) e. ran F /\ A.z e. ran F -. zEy) -> -. (F` suc x) e. (F` x)))
32		fnfvelrn 3813	. . . . . . . . . . . 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 288	. . . . . . . . . 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 3150	. . . . . . . . 9 |- (x e. om -> suc x e. om)
38	36, 37	sylan2i 465	. . . . . . . 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 1739	. . . . . 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 203	. . . . 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 373	. . . 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 1746	. 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 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280   class class class wbr 2619  Ecep 2830   Fr wfr 2915  suc csuc 2950  omcom 3131  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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