Theorem tfr2 2963

Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function F has the property that for any function G whatsoever, the "next" value of F is G recursively applied to all "previous" values of F.

Hypotheses

Ref	Expression
tfr.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
tfr.2	⊢ F = ∪A

Assertion

Ref	Expression
tfr2	⊢ (z ∈ On → (F ‘z) = (G ‘(F ↾ z)))

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

Proof of Theorem tfr2

Step	Hyp	Ref	Expression
1		fveq2 2832	. . 3 ⊢ (y = z → (F ‘y) = (F ‘z))
2		reseq2 2576	. . . 4 ⊢ (y = z → (F ↾ y) = (F ↾ z))
3	2	fveq2d 2836	. . 3 ⊢ (y = z → (G ‘(F ↾ y)) = (G ‘(F ↾ z)))
4	1, 3	cleq12d 1115	. 2 ⊢ (y = z → ((F ‘y) = (G ‘(F ↾ y)) ↔ (F ‘z) = (G ‘(F ↾ z))))
5		tfr.1	. . . . 5 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
6		tfr.2	. . . . 5 ⊢ F = ∪A
7		cleqid 1102	. . . . 5 ⊢ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}) = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
8	5, 6, 7	tfrlem13 2961	. . . 4 ⊢ dom F = On
9	8	eleq2i 1153	. . 3 ⊢ (y ∈ dom F ↔ y ∈ On)
10	5, 6	tfrlem9 2957	. . 3 ⊢ (y ∈ dom F → (F ‘y) = (G ‘(F ↾ y)))
11	9, 10	sylbir 176	. 2 ⊢ (y ∈ On → (F ‘y) = (G ‘(F ↾ y)))
12	4, 11	vtoclga 1387	1 ⊢ (z ∈ On → (F ‘z) = (G ‘(F ↾ z)))

Syntax hints:   → wi 2   ∧ wa 196   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∪ cun 1485  {csn 1808  ⟨cop 1810  ∪cuni 1919  Oncon0 2199  dom cdm 2410   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfr3 2964  rdgval 2978  numthlem 3598  zornlem1 3603

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-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-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

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