Theorem hbrdg 3933

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 3929	. 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-17 970	. . . . 5 |- (w e. On -> A.x w e. On)
3		ax-17 970	. . . . . 6 |- (f Fn w -> A.x f Fn w)
4		ax-17 970	. . . . . . 7 |- (v e. w -> A.x v e. w)
5		ax-17 970	. . . . . . . 8 |- (y e. (f` v) -> A.x y e. (f` v))
6		ax-17 970	. . . . . . . . . . 11 |- (y e. z -> A.x y e. z)
7		ax-17 970	. . . . . . . . . . . 12 |- (g = (/) -> A.x g = (/))
8		hbrdg.2	. . . . . . . . . . . 12 |- (y e. A -> A.x y e. A)
9		ax-17 970	. . . . . . . . . . . . 13 |- (Lim dom g -> A.xLim dom g)
10		ax-17 970	. . . . . . . . . . . . 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-17 970	. . . . . . . . . . . . . 14 |- (y e. (g` U.dom g) -> A.x y e. (g` U.dom g))
13	11, 12	hbfv 3726	. . . . . . . . . . . . 13 |- (y e. (F` (g` U.dom g)) -> A.x y e. (F` (g` U.dom g)))
14	9, 10, 13	hbif 2371	. . . . . . . . . . . 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 2371	. . . . . . . . . . 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 1564	. . . . . . . . . 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 2809	. . . . . . . . 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-17 970	. . . . . . . . 9 |- (y e. (f |` v) -> A.x y e. (f |` v))
19	17, 18	hbfv 3726	. . . . . . . 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 1564	. . . . . . 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 1685	. . . . . 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 1008	. . . . 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 1687	. . . 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 1467	. . 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 2506	. 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 1562	1 |- (y e. rec(F, A) -> A.x y e. rec(F, A))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  {cab 1463  A.wral 1644  E.wrex 1645  (/)c0 2278  ifcif 2359  U.cuni 2500  {copab 2663  Oncon0 2945  Lim wlim 2946  dom cdm 3167  ran crn 3168   |` cres 3169   Fn wfn 3174  ` cfv 3179  reccrdg 3928

This theorem is referenced by:  rdgsucopab 3943  rdgsucopabn 3944  frsucopab 3951  abianfplem 3958  unbnn 4534  inf0 4593  trcl 4632  alephfplem2 4884  om2uzsuc 6251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-xp 3181  df-cnv 3183  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fv 3195  df-rdg 3929

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