Theorem tfr2 3910

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 3709	. . 3 ⊢ (y = z → (F ‘y) = (F ‘z))
2		reseq2 3353	. . . 4 ⊢ (y = z → (F ↾ y) = (F ↾ z))
3	2	fveq2d 3713	. . 3 ⊢ (y = z → (G ‘(F ↾ y)) = (G ‘(F ↾ z)))
4	1, 3	eqeq12d 1481	. 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 1468	. . . . 5 ⊢ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}) = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
8	5, 6, 7	tfrlem13 3908	. . . 4 ⊢ dom F = On
9	8	eleq2i 1530	. . 3 ⊢ (y ∈ dom F ↔ y ∈ On)
10	5, 6	tfrlem9 3904	. . 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 1843	1 ⊢ (z ∈ On → (F ‘z) = (G ‘(F ↾ z)))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955  {cab 1456  ∀wral 1637  ∃wrex 1638   ∪ cun 2035  {csn 2399  ⟨cop 2401  ∪cuni 2493  Oncon0 2938  dom cdm 3160   ↾ cres 3162   Fn wfn 3167   ‘cfv 3172

This theorem is referenced by:  tfr3 3911  rdgval 3925  numthlem 4755  zorn2lem1 4760

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 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

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 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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