Theorem tfri2 6256

Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6255). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)

Hypotheses

Ref	Expression
tfri1.1	⊢ 𝐹 = recs(𝐺)
tfri1.2	⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)

Assertion

Ref	Expression
tfri2	⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Distinct variable group:   𝑥,𝐺

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem tfri2

Step	Hyp	Ref	Expression
1		tfri1.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
2		tfri1.2	. . . . 5 ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
3	1, 2	tfri1 6255	. . . 4 ⊢ 𝐹 Fn On
4		fndm 5217	. . . 4 ⊢ (𝐹 Fn On → dom 𝐹 = On)
5	3, 4	ax-mp 5	. . 3 ⊢ dom 𝐹 = On
6	5	eleq2i 2204	. 2 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ On)
7	1	tfr2a 6211	. 2 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
8	6, 7	sylbir 134	1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2681  Oncon0 4280  dom cdm 4534   ↾ cres 4536  Fun wfun 5112   Fn wfn 5113  ‘cfv 5118  recscrecs 6194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-recs 6195

This theorem is referenced by:  tfri3  6257