Theorem tfr2 3905

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 groups:   x,y,f,A   x,F,y,f   x,G,y,f   y,z

Proof of Theorem tfr2

Step	Hyp	Ref	Expression
1		fveq2 3704	. . 3 ⊢ (y = z → (F ‘y) = (F ‘z))
2		reseq2 3349	. . . 4 ⊢ (y = z → (F ↾ y) = (F ↾ z))
3	2	fveq2d 3708	. . 3 ⊢ (y = z → (G ‘(F ↾ y)) = (G ‘(F ↾ z)))
4	1, 3	eqeq12d 1480	. 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		eqid 1467	. . . . 5 ⊢ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}) = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
8	5, 6, 7	tfrlem13 3903	. . . 4 ⊢ dom F = On
9	8	eleq2i 1529	. . 3 ⊢ (y ∈ dom F ↔ y ∈ On)
10	5, 6	tfrlem9 3899	. . 3 ⊢ (y ∈ dom F → (F ‘y) = (G ‘(F ↾ y)))
11	9, 10	sylbir 201	. 2 ⊢ (y ∈ On → (F ‘y) = (G ‘(F ↾ y)))
12	4, 11	vtoclga 1842	1 ⊢ (z ∈ On → (F ‘z) = (G ‘(F ↾ z)))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955  {cab 1455  ∀wral 1636  ∃wrex 1637   ∪ cun 2034  {csn 2398  ⟨cop 2400  ∪cuni 2492  Oncon0 2935  dom cdm 3157   ↾ cres 3159   Fn wfn 3164   ‘cfv 3169

This theorem is referenced by:  tfr3 3906  rdgval 3920  numthlem 4750  zorn2lem1 4755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1451  ax-rep 2682  ax-sep 2692  ax-nul 2699  ax-pow 2731  ax-pr 2768  ax-un 2854

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1374  df-mo 1375  df-clab 1456  df-cleq 1461  df-clel 1464  df-ne 1578  df-ral 1640  df-rex 1641  df-rab 1643  df-v 1802  df-sbc 1931  df-dif 2038  df-un 2039  df-in 2040  df-ss 2042  df-nul 2270  df-pw 2391  df-sn 2401  df-pr 2402  df-tp 2404  df-op 2405  df-uni 2493  df-iun 2557  df-br 2609  df-opab 2656  df-tr 2670  df-eprel 2818  df-id 2821  df-po 2828  df-so 2838  df-fr 2904  df-we 2921  df-ord 2938  df-on 2939  df-suc 2941  df-xp 3171  df-rel 3172  df-cnv 3173  df-co 3174  df-dm 3175  df-rn 3176  df-res 3177  df-ima 3178  df-fun 3179  df-fn 3180  df-fv 3185

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