Theorem rdgsucopabn 5155

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with rdgsucopab 5154 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 groups:   z,D   y,z,C   z,A   z,B   x,y,z

Proof of Theorem rdgsucopabn

Step	Hyp	Ref	Expression
1		rdgsuc 5153	. . . . 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 4682	. . . . 5 |- (F` suc B) = (rec({<.x, y>. | y = C}, A)` suc B)
4	1, 3	syl5eq 1940	. . . 4 |- (B e. On -> (F` suc B) = ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)))
5		hbopab1 3562	. . . . . . 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 5144	. . . . . 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 4686	. . . . 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 4682	. . . . . . 7 |- (F` B) = (rec({<.x, y>. | y = C}, A)` B)
12	11	eqeq2i 1894	. . . . . 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 218	. . . . 5 |- (x = (rec({<.x, y>. | y = C}, A)` B) -> C = D)
15	9, 10, 14	fvopabnf 4751	. . . 4 |- (-. D e. _V -> ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)) = (/))
16	4, 15	sylan9eq 1948	. . 3 |- ((B e. On /\ -. D e. _V) -> (F` suc B) = (/))
17	16	ex 402	. 2 |- (B e. On -> (-. D e. _V -> (F` suc B) = (/)))
18		sucelon 3898	. . . . . 6 |- (B e. On <-> suc B e. On)
19	2	dmeqi 4158	. . . . . . . 8 |- dom F = dom rec({<.x, y>. | y = C}, A)
20		rdgfnon 5147	. . . . . . . . 9 |- rec({<.x, y>. | y = C}, A) Fn On
21		fndm 4512	. . . . . . . . 9 |- (rec({<.x, y>. | y = C}, A) Fn On -> dom rec({<.x, y>. | y = C}, A) = On)
22	20, 21	ax-mp 7	. . . . . . . 8 |- dom rec({<.x, y>. | y = C}, A) = On
23	19, 22	eqtri 1908	. . . . . . 7 |- dom F = On
24	23	eleq2i 1961	. . . . . 6 |- (suc B e. dom F <-> suc B e. On)
25	18, 24	bitr4i 193	. . . . 5 |- (B e. On <-> suc B e. dom F)
26	25	notbii 204	. . . 4 |- (-. B e. On <-> -. suc B e. dom F)
27		ndmfv 4702	. . . 4 |- (-. suc B e. dom F -> (F` suc B) = (/))
28	26, 27	sylbi 216	. . 3 |- (-. B e. On -> (F` suc B) = (/))
29	28	a1d 15	. 2 |- (-. B e. On -> (-. D e. _V -> (F` suc B) = (/)))
30	17, 29	pm2.61i 140	1 |- (-. D e. _V -> (F` suc B) = (/))

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {copab 3395  Oncon0 3657  suc csuc 3659  dom cdm 3986   Fn wfn 3993  ` cfv 3998  reccrdg 5139

This theorem is referenced by:  alephon 5876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

Copyright terms: Public domain