Theorem hbrdg 2974

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

Hypotheses

Ref	Expression
hbrdg.1	⊢ (y ∈ F → ∀x y ∈ F)
hbrdg.2	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbrdg	⊢ (y ∈ rec(F, A) → ∀x y ∈ rec(F, A))

Distinct variable group(s):   y,F   y,A   x,y

Proof of Theorem hbrdg

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (w ∈ On → ∀x w ∈ On)
2		ax-17 925	. . . . . 6 ⊢ (f Fn w → ∀x f Fn w)
3		ax-17 925	. . . . . . 7 ⊢ (v ∈ w → ∀x v ∈ w)
4		ax-17 925	. . . . . . . 8 ⊢ (y ∈ (f ‘v) → ∀x y ∈ (f ‘v))
5		ax-17 925	. . . . . . . . . . . 12 ⊢ (g = ∅ → ∀x g = ∅)
6		ax-17 925	. . . . . . . . . . . . 13 ⊢ (y ∈ z → ∀x y ∈ z)
7		hbrdg.2	. . . . . . . . . . . . 13 ⊢ (y ∈ A → ∀x y ∈ A)
8	6, 7	hbeq 1171	. . . . . . . . . . . 12 ⊢ (z = A → ∀x z = A)
9	5, 8	hban 704	. . . . . . . . . . 11 ⊢ ((g = ∅ ∧ z = A) → ∀x(g = ∅ ∧ z = A))
10		ax-17 925	. . . . . . . . . . . 12 ⊢ (¬ (g = ∅ ∨ Lim dom g) → ∀x ¬ (g = ∅ ∨ Lim dom g))
11		hbrdg.1	. . . . . . . . . . . . . 14 ⊢ (y ∈ F → ∀x y ∈ F)
12		ax-17 925	. . . . . . . . . . . . . 14 ⊢ (y ∈ (g ‘∪dom g) → ∀x y ∈ (g ‘∪dom g))
13	11, 12	hbfv 2837	. . . . . . . . . . . . 13 ⊢ (y ∈ (F ‘(g ‘∪dom g)) → ∀x y ∈ (F ‘(g ‘∪dom g)))
14	6, 13	hbeq 1171	. . . . . . . . . . . 12 ⊢ (z = (F ‘(g ‘∪dom g)) → ∀x z = (F ‘(g ‘∪dom g)))
15	10, 14	hban 704	. . . . . . . . . . 11 ⊢ ((¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) → ∀x(¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))))
16		ax-17 925	. . . . . . . . . . 11 ⊢ ((Lim dom g ∧ z = ∪ran g) → ∀x(Lim dom g ∧ z = ∪ran g))
17	9, 15, 16	hb3or 706	. . . . . . . . . 10 ⊢ (((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g)) → ∀x((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g)))
18	17	hbopab 2111	. . . . . . . . 9 ⊢ (y ∈ {⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} → ∀x y ∈ {⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))})
19		ax-17 925	. . . . . . . . 9 ⊢ (y ∈ (f ↾ v) → ∀x y ∈ (f ↾ v))
20	18, 19	hbfv 2837	. . . . . . . 8 ⊢ (y ∈ ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)) → ∀x y ∈ ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))
21	4, 20	hbeq 1171	. . . . . . 7 ⊢ ((f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)) → ∀x(f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))
22	3, 21	hbral 1236	. . . . . 6 ⊢ (∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)) → ∀x∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))
23	2, 22	hban 704	. . . . 5 ⊢ ((f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v))) → ∀x(f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v))))
24	1, 23	hbrex 1238	. . . 4 ⊢ (∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v))) → ∀x∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v))))
25	24	hbab 1096	. . 3 ⊢ (y ∈ {f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))} → ∀x y ∈ {f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))})
26	25	hbuni 1925	. 2 ⊢ (y ∈ ∪{f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))} → ∀x y ∈ ∪{f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))})
27		df-rdg 2970	. . 3 ⊢ rec(F, A) = ∪{f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))}
28	27	eleq2i 1153	. 2 ⊢ (y ∈ rec(F, A) ↔ y ∈ ∪{f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))})
29	28	bial 695	. 2 ⊢ (∀x y ∈ rec(F, A) ↔ ∀x y ∈ ∪{f∣∃w ∈ On (f Fn w ∧ ∀v ∈ w (f ‘v) = ({⟨g, z⟩∣((g = ∅ ∧ z = A) ∨ (¬ (g = ∅ ∨ Lim dom g) ∧ z = (F ‘(g ‘∪dom g))) ∨ (Lim dom g ∧ z = ∪ran g))} ‘(f ↾ v)))})
30	26, 28, 29	3imtr4 192	1 ⊢ (y ∈ rec(F, A) → ∀x y ∈ rec(F, A))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∨ w3o 580  ∀wal 672   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  rdgsucopab 2984  rdgsucopabn 2985  frsucopab 2992  abianfplem 2999  unbnn 3435  inf0 3457  trcl 3489  om2uzsuc 4652

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-rdg 2970

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