Theorem tfr2 3916

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 3715	. . 3 ⊢ (y = z → (F ‘y) = (F ‘z))
2		reseq2 3361	. . . 4 ⊢ (y = z → (F ↾ y) = (F ↾ z))
3	2	fveq2d 3719	. . 3 ⊢ (y = z → (G ‘(F ↾ y)) = (G ‘(F ↾ z)))
4	1, 3	eqeq12d 1486	. 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 1473	. . . . 5 ⊢ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}) = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
8	5, 6, 7	tfrlem13 3914	. . . 4 ⊢ dom F = On
9	8	eleq2i 1535	. . 3 ⊢ (y ∈ dom F ↔ y ∈ On)
10	5, 6	tfrlem9 3910	. . 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 1848	1 ⊢ (z ∈ On → (F ‘z) = (G ‘(F ↾ z)))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 954   ∈ wcel 956  {cab 1461  ∀wral 1642  ∃wrex 1643   ∪ cun 2041  {csn 2405  ⟨cop 2407  ∪cuni 2498  Oncon0 2943  dom cdm 3165   ↾ cres 3167   Fn wfn 3172   ‘cfv 3177

This theorem is referenced by:  tfr3 3917  rdgval 3931  numthlem 4763  zorn2lem1 4768

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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