Theorem inf3lem1 4613

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4620 for detailed description.

Hypotheses

Ref	Expression
inf3lem.1	|- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
inf3lem.2	|- F = (rec(G, (/)) |` om)
inf3lem.3	|- A e. V
inf3lem.4	|- B e. V

Assertion

Ref	Expression
inf3lem1	|- (A e. om -> (F` A) (_ (F` suc A))

Distinct variable group:   x,y,z,w

Proof of Theorem inf3lem1

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (v = (/) -> (F` v) = (F` (/)))
2		suceq 3034	. . . 4 |- (v = (/) -> suc v = suc (/))
3	2	fveq2d 3728	. . 3 |- (v = (/) -> (F` suc v) = (F` suc (/)))
4	1, 3	sseq12d 2090	. 2 |- (v = (/) -> ((F` v) (_ (F` suc v) <-> (F` (/)) (_ (F` suc (/))))
5		fveq2 3724	. . 3 |- (v = u -> (F` v) = (F` u))
6		suceq 3034	. . . 4 |- (v = u -> suc v = suc u)
7	6	fveq2d 3728	. . 3 |- (v = u -> (F` suc v) = (F` suc u))
8	5, 7	sseq12d 2090	. 2 |- (v = u -> ((F` v) (_ (F` suc v) <-> (F` u) (_ (F` suc u)))
9		fveq2 3724	. . 3 |- (v = suc u -> (F` v) = (F` suc u))
10		suceq 3034	. . . 4 |- (v = suc u -> suc v = suc suc u)
11	10	fveq2d 3728	. . 3 |- (v = suc u -> (F` suc v) = (F` suc suc u))
12	9, 11	sseq12d 2090	. 2 |- (v = suc u -> ((F` v) (_ (F` suc v) <-> (F` suc u) (_ (F` suc suc u)))
13		fveq2 3724	. . 3 |- (v = A -> (F` v) = (F` A))
14		suceq 3034	. . . 4 |- (v = A -> suc v = suc A)
15	14	fveq2d 3728	. . 3 |- (v = A -> (F` suc v) = (F` suc A))
16	13, 15	sseq12d 2090	. 2 |- (v = A -> ((F` v) (_ (F` suc v) <-> (F` A) (_ (F` suc A)))
17		inf3lem.1	. . . 4 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
18		inf3lem.2	. . . 4 |- F = (rec(G, (/)) |` om)
19		inf3lem.3	. . . 4 |- A e. V
20	17, 18, 19, 19	inf3lemb 4610	. . 3 |- (F` (/)) = (/)
21		0ss 2301	. . 3 |- (/) (_ (F` suc (/))
22	20, 21	eqsstr 2091	. 2 |- (F` (/)) (_ (F` suc (/))
23		visset 1813	. . . . . . . . . 10 |- u e. V
24	17, 18, 23, 23	inf3lemc 4611	. . . . . . . . 9 |- (u e. om -> (F` suc u) = (G` (F` u)))
25	24	eleq2d 1541	. . . . . . . 8 |- (u e. om -> (v e. (F` suc u) <-> v e. (G` (F` u))))
26		visset 1813	. . . . . . . . 9 |- v e. V
27		fvex 3732	. . . . . . . . 9 |- (F` u) e. V
28	17, 18, 26, 27	inf3lema 4609	. . . . . . . 8 |- (v e. (G` (F` u)) <-> (v e. x /\ (v i^i x) (_ (F` u)))
29	25, 28	syl6bb 536	. . . . . . 7 |- (u e. om -> (v e. (F` suc u) <-> (v e. x /\ (v i^i x) (_ (F` u))))
30		peano2b 3147	. . . . . . . . . 10 |- (u e. om <-> suc u e. om)
31	23	sucex 3050	. . . . . . . . . . 11 |- suc u e. V
32	17, 18, 31, 23	inf3lemc 4611	. . . . . . . . . 10 |- (suc u e. om -> (F` suc suc u) = (G` (F` suc u)))
33	30, 32	sylbi 199	. . . . . . . . 9 |- (u e. om -> (F` suc suc u) = (G` (F` suc u)))
34	33	eleq2d 1541	. . . . . . . 8 |- (u e. om -> (v e. (F` suc suc u) <-> v e. (G` (F` suc u))))
35		fvex 3732	. . . . . . . . 9 |- (F` suc u) e. V
36	17, 18, 26, 35	inf3lema 4609	. . . . . . . 8 |- (v e. (G` (F` suc u)) <-> (v e. x /\ (v i^i x) (_ (F` suc u)))
37	34, 36	syl6bb 536	. . . . . . 7 |- (u e. om -> (v e. (F` suc suc u) <-> (v e. x /\ (v i^i x) (_ (F` suc u))))
38	29, 37	imbi12d 626	. . . . . 6 |- (u e. om -> ((v e. (F` suc u) -> v e. (F` suc suc u)) <-> ((v e. x /\ (v i^i x) (_ (F` u)) -> (v e. x /\ (v i^i x) (_ (F` suc u)))))
39		sstr2 2071	. . . . . . . 8 |- ((v i^i x) (_ (F` u) -> ((F` u) (_ (F` suc u) -> (v i^i x) (_ (F` suc u)))
40	39	com12 11	. . . . . . 7 |- ((F` u) (_ (F` suc u) -> ((v i^i x) (_ (F` u) -> (v i^i x) (_ (F` suc u)))
41	40	anim2d 561	. . . . . 6 |- ((F` u) (_ (F` suc u) -> ((v e. x /\ (v i^i x) (_ (F` u)) -> (v e. x /\ (v i^i x) (_ (F` suc u))))
42	38, 41	syl5bir 210	. . . . 5 |- (u e. om -> ((F` u) (_ (F` suc u) -> (v e. (F` suc u) -> v e. (F` suc suc u))))
43	42	imp 350	. . . 4 |- ((u e. om /\ (F` u) (_ (F` suc u)) -> (v e. (F` suc u) -> v e. (F` suc suc u)))
44	43	ssrdv 2070	. . 3 |- ((u e. om /\ (F` u) (_ (F` suc u)) -> (F` suc u) (_ (F` suc suc u))
45	44	ex 373	. 2 |- (u e. om -> ((F` u) (_ (F` suc u) -> (F` suc u) (_ (F` suc suc u)))
46	4, 8, 12, 16, 22, 45	finds 3156	1 |- (A e. om -> (F` A) (_ (F` suc A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  {copab 2666  suc csuc 2950  omcom 3131   |` cres 3172  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  inf3lem4 4616

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-9 965  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

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-rab 1652  df-v 1812  df-sbc 1942  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-iun 2568  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  df-rdg 3932

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