Theorem tfri2d 6339

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 6368). 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
tfri1d.1	⊢ 𝐹 = recs(𝐺)
tfri1d.2	⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))

Assertion

Ref	Expression
tfri2d	⊢ ((𝜑 ∧ 𝐴 ∈ On) → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Distinct variable group:   𝑥,𝐺

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

Proof of Theorem tfri2d

Step	Hyp	Ref	Expression
1		tfri1d.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
2		tfri1d.2	. . . . . 6 ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
3	1, 2	tfri1d 6338	. . . . 5 ⊢ (𝜑 → 𝐹 Fn On)
4		fndm 5317	. . . . 5 ⊢ (𝐹 Fn On → dom 𝐹 = On)
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → dom 𝐹 = On)
6	5	eleq2d 2247	. . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ On))
7	6	biimpar 297	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ On) → 𝐴 ∈ dom 𝐹)
8	1	tfr2a 6324	. 2 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
9	7, 8	syl 14	1 ⊢ ((𝜑 ∧ 𝐴 ∈ On) → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351   = wceq 1353   ∈ wcel 2148  Vcvv 2739  Oncon0 4365  dom cdm 4628   ↾ cres 4630  Fun wfun 5212   Fn wfn 5213  ‘cfv 5218  recscrecs 6307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308

This theorem is referenced by:  rdgivallem  6384