Theorem rdgsucopabn 2985

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered pair abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with rdgsucopab 2984 to help eliminate redundant sethood antecedents.

Hypotheses

Ref	Expression
rdgsucopab.1	|- (z e. A -> A.x z e. A)
rdgsucopab.2	|- (z e. B -> A.x z e. B)
rdgsucopab.3	|- (z e. D -> A.x z e. D)
rdgsucopab.4	|- F = rec({<.x, y>. | y = C}, A)
rdgsucopab.5	|- (x = (F` B) -> C = D)

Assertion

Ref	Expression
rdgsucopabn	|- (-. D e. V -> (F` suc B) = (/))

Distinct variable group(s):   z,D   y,z,C   z,A   z,B   x,y,z

Proof of Theorem rdgsucopabn

Step	Hyp	Ref	Expression
1		rdgsuct 2983	. . . . 5 |- (B e. On -> (rec({<.x, y>. | y = C}, A)` suc B) = ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)))
2		rdgsucopab.4	. . . . . 6 |- F = rec({<.x, y>. | y = C}, A)
3	2	fveq1i 2833	. . . . 5 |- (F` suc B) = (rec({<.x, y>. | y = C}, A)` suc B)
4	1, 3	syl5eq 1136	. . . 4 |- (B e. On -> (F` suc B) = ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)))
5		hbopab1 2112	. . . . . . 7 |- (z e. {<.x, y>. | y = C} -> A.x z e. {<.x, y>. | y = C})
6		rdgsucopab.1	. . . . . . 7 |- (z e. A -> A.x z e. A)
7	5, 6	hbrdg 2974	. . . . . 6 |- (z e. rec({<.x, y>. | y = C}, A) -> A.x z e. rec({<.x, y>. | y = C}, A))
8		rdgsucopab.2	. . . . . 6 |- (z e. B -> A.x z e. B)
9	7, 8	hbfv 2837	. . . . 5 |- (z e. (rec({<.x, y>. | y = C}, A)` B) -> A.x z e. (rec({<.x, y>. | y = C}, A)` B))
10		rdgsucopab.3	. . . . 5 |- (z e. D -> A.x z e. D)
11	2	fveq1i 2833	. . . . . . 7 |- (F` B) = (rec({<.x, y>. | y = C}, A)` B)
12	11	cleq2i 1111	. . . . . 6 |- (x = (F` B) <-> x = (rec({<.x, y>. | y = C}, A)` B))
13		rdgsucopab.5	. . . . . 6 |- (x = (F` B) -> C = D)
14	12, 13	sylbir 176	. . . . 5 |- (x = (rec({<.x, y>. | y = C}, A)` B) -> C = D)
15	9, 10, 14	fvopabnf 2875	. . . 4 |- (-. D e. V -> ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)) = (/))
16	4, 15	sylan9eq 1144	. . 3 |- ((B e. On /\ -. D e. V) -> (F` suc B) = (/))
17	16	exp 291	. 2 |- (B e. On -> (-. D e. V -> (F` suc B) = (/)))
18		sucelon 2319	. . . . . 6 |- (B e. On <-> suc B e. On)
19	2	dmeqi 2532	. . . . . . . 8 |- dom F = dom rec({<.x, y>. | y = C}, A)
20		rdgfnon 2977	. . . . . . . . 9 |- rec({<.x, y>. | y = C}, A) Fn On
21		fndm 2723	. . . . . . . . 9 |- (rec({<.x, y>. | y = C}, A) Fn On -> dom rec({<.x, y>. | y = C}, A) = On)
22	20, 21	ax-mp 6	. . . . . . . 8 |- dom rec({<.x, y>. | y = C}, A) = On
23	19, 22	eqtr 1119	. . . . . . 7 |- dom F = On
24	23	eleq2i 1153	. . . . . 6 |- (suc B e. dom F <-> suc B e. On)
25	18, 24	bitr4 154	. . . . 5 |- (B e. On <-> suc B e. dom F)
26	25	negbii 162	. . . 4 |- (-. B e. On <-> -. suc B e. dom F)
27		ndmfv 2848	. . . 4 |- (-. suc B e. dom F -> (F` suc B) = (/))
28	26, 27	sylbi 174	. . 3 |- (-. B e. On -> (F` suc B) = (/))
29	28	a1d 14	. 2 |- (-. B e. On -> (-. D e. V -> (F` suc B) = (/)))
30	17, 29	pm2.61i 110	1 |- (-. D e. V -> (F` suc B) = (/))

Syntax hints:  -. wn 1   -> wi 2  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {copab 2055  Oncon0 2199  suc csuc 2201  dom cdm 2410   Fn wfn 2417  ` cfv 2422  reccrdg 2969

This theorem is referenced by:  alephon 3671

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

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