Theorem hbrdg 4237

Description: Bound-variable hypothesis builder for the recursive definition generator.

Hypotheses

Ref	Expression
hbrdg.1	|- (y e. F -> A.x y e. F)
hbrdg.2	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbrdg	|- (y e. rec(F, A) -> A.x y e. rec(F, A))

Distinct variable groups:   y,F   y,A   x,y

Proof of Theorem hbrdg

Step	Hyp	Ref	Expression
1		df-rdg 4233	. 2 |- rec(F, A) = U.{f | E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))}
2		ax-17 1007	. . . . 5 |- (w e. On -> A.x w e. On)
3		ax-17 1007	. . . . . 6 |- (f Fn w -> A.x f Fn w)
4		ax-17 1007	. . . . . . 7 |- (v e. w -> A.x v e. w)
5		ax-17 1007	. . . . . . . 8 |- (y e. (f` v) -> A.x y e. (f` v))
6		ax-17 1007	. . . . . . . . . . 11 |- (y e. z -> A.x y e. z)
7		ax-17 1007	. . . . . . . . . . . 12 |- (g = (/) -> A.x g = (/))
8		hbrdg.2	. . . . . . . . . . . 12 |- (y e. A -> A.x y e. A)
9		ax-17 1007	. . . . . . . . . . . . 13 |- (Lim dom g -> A.xLim dom g)
10		ax-17 1007	. . . . . . . . . . . . 13 |- (y e. U.ran g -> A.x y e. U.ran g)
11		hbrdg.1	. . . . . . . . . . . . . 14 |- (y e. F -> A.x y e. F)
12		ax-17 1007	. . . . . . . . . . . . . 14 |- (y e. (g` U.dom g) -> A.x y e. (g` U.dom g))
13	11, 12	hbfv 3840	. . . . . . . . . . . . 13 |- (y e. (F` (g` U.dom g)) -> A.x y e. (F` (g` U.dom g)))
14	9, 10, 13	hbif 2427	. . . . . . . . . . . 12 |- (y e. if(Lim dom g, U.ran g, (F` (g` U.dom g))) -> A.x y e. if(Lim dom g, U.ran g, (F` (g` U.dom g))))
15	7, 8, 14	hbif 2427	. . . . . . . . . . 11 |- (y e. if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))) -> A.x y e. if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))))
16	6, 15	hbeq 1608	. . . . . . . . . 10 |- (z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))) -> A.x z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))))
17	16	hbopab 2889	. . . . . . . . 9 |- (y e. {<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))} -> A.x y e. {<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))})
18		ax-17 1007	. . . . . . . . 9 |- (y e. (f |` v) -> A.x y e. (f |` v))
19	17, 18	hbfv 3840	. . . . . . . 8 |- (y e. ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)) -> A.x y e. ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))
20	5, 19	hbeq 1608	. . . . . . 7 |- ((f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)) -> A.x(f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))
21	4, 20	hbral 1732	. . . . . 6 |- (A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)) -> A.xA.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))
22	3, 21	hban 1045	. . . . 5 |- ((f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v))) -> A.x(f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v))))
23	2, 22	hbrex 1734	. . . 4 |- (E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v))) -> A.xE.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v))))
24	23	hbab 1509	. . 3 |- (y e. {f | E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))} -> A.x y e. {f | E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))})
25	24	hbuni 2575	. 2 |- (y e. U.{f | E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))} -> A.x y e. U.{f | E.w e. On (f Fn w /\ A.v e. w (f` v) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` v)))})
26	1, 25	hbxfr 1606	1 |- (y e. rec(F, A) -> A.x y e. rec(F, A))

Syntax hints:   -> wi 3   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  {cab 1505  A.wral 1691  E.wrex 1692  (/)c0 2332  ifcif 2415  U.cuni 2569  {copab 2740  Oncon0 2975  Lim wlim 2976  dom cdm 3251  ran crn 3252   |` cres 3253   Fn wfn 3258  ` cfv 3263  reccrdg 4232

This theorem is referenced by:  rdgsucopab 4247  rdgsucopabn 4248  frsucopab 4255  abianfplem 4262  unbnn 4690  inf0 4751  trcl 4791  alephfplem2 5047  om2uzsuci 6659  neibastop2lem1 11580  neibastop2lem3 11582  neibastop2lem4 11583

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-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  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-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-xp 3265  df-cnv 3267  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fv 3279  df-rdg 4233

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