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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtfr0dm 6301 Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
𝐹 = recs(𝐺)       ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)
 
Theoremtfr0 6302 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
𝐹 = recs(𝐺)       ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅))
 
Theoremtfrlemisucfn 6303* We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 2-Jul-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   (𝜑𝑧 ∈ On)    &   (𝜑𝑔 Fn 𝑧)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
 
Theoremtfrlemisucaccv 6304* We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   (𝜑𝑧 ∈ On)    &   (𝜑𝑔 Fn 𝑧)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
 
Theoremtfrlemibacc 6305* Each element of 𝐵 is an acceptable function. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑𝐵𝐴)
 
Theoremtfrlemibxssdm 6306* The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑𝑥 ⊆ dom 𝐵)
 
Theoremtfrlemibfn 6307* The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑 𝐵 Fn 𝑥)
 
Theoremtfrlemibex 6308* The set 𝐵 exists. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑𝐵 ∈ V)
 
Theoremtfrlemiubacc 6309* The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6311). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
 
Theoremtfrlemiex 6310* Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))    &   𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}    &   (𝜑𝑥 ∈ On)    &   (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))       (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
 
Theoremtfrlemi1 6311* We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that 𝐹 is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))       ((𝜑𝐶 ∈ On) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
 
Theoremtfrlemi14d 6312* The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))       (𝜑 → dom recs(𝐹) = On)
 
Theoremtfrexlem 6313* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))       ((𝜑𝐶𝑉) → (recs(𝐹)‘𝐶) ∈ V)
 
Theoremtfri1d 6314* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that 𝐺 is defined "everywhere", which is stated here as (𝐺𝑥) ∈ V. Alternately, 𝑥 ∈ On∀𝑓(𝑓 Fn 𝑥𝑓 ∈ dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

𝐹 = recs(𝐺)    &   (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))       (𝜑𝐹 Fn On)
 
Theoremtfri2d 6315* 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 6344). 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.)
𝐹 = recs(𝐺)    &   (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))       ((𝜑𝐴 ∈ On) → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
 
Theoremtfr1onlem3ag 6316* Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6288 but for tfr1on 6329 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.)
𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}       (𝐻𝑉 → (𝐻𝐴 ↔ ∃𝑧𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐻𝑤) = (𝐺‘(𝐻𝑤)))))
 
Theoremtfr1onlem3 6317* Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3 6290 but for tfr1on 6329 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)
𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}       𝐴 = {𝑔 ∣ ∃𝑧𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}
 
Theoremtfr1onlemssrecs 6318* Lemma for tfr1on 6329. The union of functions acceptable for tfr1on 6329 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)
𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑 → Ord 𝑋)       (𝜑 𝐴 ⊆ recs(𝐺))
 
Theoremtfr1onlemsucfn 6319* We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6329. (Contributed by Jim Kingdon, 12-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑𝑧𝑋)    &   (𝜑𝑔 Fn 𝑧)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
 
Theoremtfr1onlemsucaccv 6320* Lemma for tfr1on 6329. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑𝑌𝑋)    &   (𝜑𝑧𝑌)    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑔 Fn 𝑧)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
 
Theoremtfr1onlembacc 6321* Lemma for tfr1on 6329. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐵𝐴)
 
Theoremtfr1onlembxssdm 6322* Lemma for tfr1on 6329. The union of 𝐵 is defined on all elements of 𝑋. (Contributed by Jim Kingdon, 14-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐷 ⊆ dom 𝐵)
 
Theoremtfr1onlembfn 6323* Lemma for tfr1on 6329. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 15-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 𝐵 Fn 𝐷)
 
Theoremtfr1onlembex 6324* Lemma for tfr1on 6329. The set 𝐵 exists. (Contributed by Jim Kingdon, 14-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐵 ∈ V)
 
Theoremtfr1onlemubacc 6325* Lemma for tfr1on 6329. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
 
Theoremtfr1onlemex 6326* Lemma for tfr1on 6329. (Contributed by Jim Kingdon, 16-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
 
Theoremtfr1onlemaccex 6327* We can define an acceptable function on any element of 𝑋.

As with many of the transfinite recursion theorems, we have hypotheses that state that 𝐹 is a function and that it is defined up to 𝑋. (Contributed by Jim Kingdon, 16-Mar-2022.)

𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)       ((𝜑𝐶𝑋) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
 
Theoremtfr1onlemres 6328* Lemma for tfr1on 6329. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑌𝑋)       (𝜑𝑌 ⊆ dom 𝐹)
 
Theoremtfr1on 6329* Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑌𝑋)       (𝜑𝑌 ⊆ dom 𝐹)
 
Theoremtfri1dALT 6330* Alternate proof of tfri1d 6314 in terms of tfr1on 6329.

Although this does show that the tfr1on 6329 proof is general enough to also prove tfri1d 6314, the tfri1d 6314 proof is simpler in places because it does not need to deal with 𝑋 being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

𝐹 = recs(𝐺)    &   (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))       (𝜑𝐹 Fn On)
 
Theoremtfrcllemssrecs 6331* Lemma for tfrcl 6343. The union of functions acceptable for tfrcl 6343 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑 → Ord 𝑋)       (𝜑 𝐴 ⊆ recs(𝐺))
 
Theoremtfrcllemsucfn 6332* We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6343. (Contributed by Jim Kingdon, 24-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑𝑧𝑋)    &   (𝜑𝑔:𝑧𝑆)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}):suc 𝑧𝑆)
 
Theoremtfrcllemsucaccv 6333* Lemma for tfrcl 6343. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   (𝜑𝑌𝑋)    &   (𝜑𝑧𝑌)    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑔:𝑧𝑆)    &   (𝜑𝑔𝐴)       (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
 
Theoremtfrcllembacc 6334* Lemma for tfrcl 6343. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐵𝐴)
 
Theoremtfrcllembxssdm 6335* Lemma for tfrcl 6343. The union of 𝐵 is defined on all elements of 𝑋. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐷 ⊆ dom 𝐵)
 
Theoremtfrcllembfn 6336* Lemma for tfrcl 6343. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 𝐵:𝐷𝑆)
 
Theoremtfrcllembex 6337* Lemma for tfrcl 6343. The set 𝐵 exists. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑𝐵 ∈ V)
 
Theoremtfrcllemubacc 6338* Lemma for tfrcl 6343. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 → ∀𝑢𝐷 ( 𝐵𝑢) = (𝐺‘( 𝐵𝑢)))
 
Theoremtfrcllemex 6339* Lemma for tfrcl 6343. (Contributed by Jim Kingdon, 26-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   𝐵 = { ∣ ∃𝑧𝐷𝑔(𝑔:𝑧𝑆𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑 → ∀𝑧𝐷𝑔(𝑔:𝑧𝑆 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤))))       (𝜑 → ∃𝑓(𝑓:𝐷𝑆 ∧ ∀𝑢𝐷 (𝑓𝑢) = (𝐺‘(𝑓𝑢))))
 
Theoremtfrcllemaccex 6340* We can define an acceptable function on any element of 𝑋.

As with many of the transfinite recursion theorems, we have hypotheses that state that 𝐹 is a function and that it is defined up to 𝑋. (Contributed by Jim Kingdon, 26-Mar-2022.)

𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)       ((𝜑𝐶𝑋) → ∃𝑔(𝑔:𝐶𝑆 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
 
Theoremtfrcllemres 6341* Lemma for tfr1on 6329. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑌𝑋)       (𝜑𝑌 ⊆ dom 𝐹)
 
Theoremtfrcldm 6342* Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑌 𝑋)       (𝜑𝑌 ∈ dom 𝐹)
 
Theoremtfrcl 6343* Closure for transfinite recursion. As with tfr1on 6329, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)
𝐹 = recs(𝐺)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → Ord 𝑋)    &   ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)    &   ((𝜑𝑥 𝑋) → suc 𝑥𝑋)    &   (𝜑𝑌 𝑋)       (𝜑 → (𝐹𝑌) ∈ 𝑆)
 
Theoremtfri1 6344* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that 𝐺 is defined "everywhere", which is stated here as (𝐺𝑥) ∈ V. Alternately, 𝑥 ∈ On∀𝑓(𝑓 Fn 𝑥𝑓 ∈ dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

𝐹 = recs(𝐺)    &   (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)       𝐹 Fn On
 
Theoremtfri2 6345* 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 6344). 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.)
𝐹 = recs(𝐺)    &   (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)       (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
 
Theoremtfri3 6346* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6344). Finally, we show that 𝐹 is unique. We do this by showing that any class 𝐵 with the same properties of 𝐹 that we showed in parts 1 and 2 is identical to 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
𝐹 = recs(𝐺)    &   (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)       ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
 
Theoremtfrex 6347* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = recs(𝐺)    &   (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))       ((𝜑𝐴𝑉) → (𝐹𝐴) ∈ V)
 
2.6.21  Recursive definition generator
 
Syntaxcrdg 6348 Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)
 
Definitiondf-irdg 6349* Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 6284 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of 𝑔 is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of 𝑔. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication.

For finite recursion we also define df-frec 6370 and for suitable characteristic functions df-frec 6370 yields the same result as rec restricted to ω, as seen at frecrdg 6387.

Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)

rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ (𝐼 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
 
Theoremrdgeq1 6350 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
 
Theoremrdgeq2 6351 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
 
Theoremrdgfun 6352 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
Fun rec(𝐹, 𝐴)
 
Theoremrdgtfr 6353* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘𝑓) ∈ V))
 
Theoremrdgruledefgg 6354* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
((𝐹 Fn V ∧ 𝐴𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘𝑓) ∈ V))
 
Theoremrdgruledefg 6355* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
𝐹 Fn V       (𝐴𝑉 → (Fun (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘𝑓) ∈ V))
 
Theoremrdgexggg 6356 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
((𝐹 Fn V ∧ 𝐴𝑉𝐵𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
 
Theoremrdgexgg 6357 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
𝐹 Fn V       ((𝐴𝑉𝐵𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
 
Theoremrdgifnon 6358 The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6365; in cases like df-oadd 6399 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
((𝐹 Fn V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
 
Theoremrdgifnon2 6359* The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
 
Theoremrdgivallem 6360* Value of the recursive definition generator. Lemma for rdgival 6361 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
 
Theoremrdgival 6361* Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
((𝐹 Fn V ∧ 𝐴𝑉𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
 
Theoremrdgss 6362 Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
(𝜑𝐹 Fn V)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐴𝐵)       (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))
 
Theoremrdgisuc1 6363* One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function 𝐹 other than 𝐹 Fn V. Given that, the resulting expression encompasses both the expected successor term (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) but also terms that correspond to the initial value 𝐴 and to limit ordinals 𝑥𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6364. (Contributed by Jim Kingdon, 9-Jun-2019.)

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ On)       (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
 
Theoremrdgisucinc 6364* Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6443 and omsuc 6451. (Contributed by Jim Kingdon, 29-Aug-2019.)

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ On)    &   (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))       (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
 
Theoremrdgon 6365* Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)       ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
 
Theoremrdg0 6366 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐴 ∈ V       (rec(𝐹, 𝐴)‘∅) = 𝐴
 
Theoremrdg0g 6367 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
(𝐴𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
 
Theoremrdgexg 6368 The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐴 ∈ V    &   𝐹 Fn V       (𝐵𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
 
2.6.22  Finite recursion
 
Syntaxcfrec 6369 Extend class notation with the finite recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class frec(𝐹, 𝐼)
 
Definitiondf-frec 6370* Define a recursive definition generator on ω (the class of finite ordinals) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6284 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6376 and frecsuc 6386.

Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4588. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6387, this definition and df-irdg 6349 restricted to ω produce the same result.

Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)

frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
 
Theoremfreceq1 6371 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))
 
Theoremfreceq2 6372 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))
 
Theoremfrecex 6373 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
frec(𝐹, 𝐴) ∈ V
 
Theoremfrecfun 6374 Finite recursion produces a function. See also frecfnom 6380 which also states that the domain of that function is ω but which puts conditions on 𝐴 and 𝐹. (Contributed by Jim Kingdon, 13-Feb-2022.)
Fun frec(𝐹, 𝐴)
 
Theoremnffrec 6375 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
𝑥𝐹    &   𝑥𝐴       𝑥frec(𝐹, 𝐴)
 
Theoremfrec0g 6376 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
(𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
 
Theoremfrecabex 6377* The class abstraction from df-frec 6370 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
(𝜑𝑆𝑉)    &   (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)    &   (𝜑𝐴𝑊)       (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ V)
 
Theoremfrecabcl 6378* The class abstraction from df-frec 6370 exists. Unlike frecabex 6377 the function 𝐹 only needs to be defined on 𝑆, not all sets. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 21-Mar-2022.)
(𝜑𝑁 ∈ ω)    &   (𝜑𝐺:𝑁𝑆)    &   (𝜑 → ∀𝑦𝑆 (𝐹𝑦) ∈ 𝑆)    &   (𝜑𝐴𝑆)       (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝐺 = suc 𝑚𝑥 ∈ (𝐹‘(𝐺𝑚))) ∨ (dom 𝐺 = ∅ ∧ 𝑥𝐴))} ∈ 𝑆)
 
Theoremfrectfr 6379* Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and 𝐴𝑉 on frec(𝐹, 𝐴), we want to be able to apply tfri1d 6314 or tfri2d 6315, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
 
Theoremfrecfnom 6380* The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)
 
Theoremfreccllem 6381* Lemma for freccl 6382. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
(𝜑𝐴𝑆)    &   ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)    &   (𝜑𝐵 ∈ ω)    &   𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))       (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
 
Theoremfreccl 6382* Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.)
(𝜑𝐴𝑆)    &   ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
 
Theoremfrecfcllem 6383* Lemma for frecfcl 6384. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))       ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
 
Theoremfrecfcl 6384* Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.)
((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
 
Theoremfrecsuclem 6385* Lemma for frecsuc 6386. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.)
𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
 
Theoremfrecsuc 6386* The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
 
Theoremfrecrdg 6387* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6370 produces the same results as df-irdg 6349 restricted to ω.

Presumably the theorem would also hold if 𝐹 Fn V were changed to 𝑧(𝐹𝑧) ∈ V. (Contributed by Jim Kingdon, 29-Aug-2019.)

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))       (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
 
2.6.23  Ordinal arithmetic
 
Syntaxc1o 6388 Extend the definition of a class to include the ordinal number 1.
class 1o
 
Syntaxc2o 6389 Extend the definition of a class to include the ordinal number 2.
class 2o
 
Syntaxc3o 6390 Extend the definition of a class to include the ordinal number 3.
class 3o
 
Syntaxc4o 6391 Extend the definition of a class to include the ordinal number 4.
class 4o
 
Syntaxcoa 6392 Extend the definition of a class to include the ordinal addition operation.
class +o
 
Syntaxcomu 6393 Extend the definition of a class to include the ordinal multiplication operation.
class ·o
 
Syntaxcoei 6394 Extend the definition of a class to include the ordinal exponentiation operation.
class o
 
Definitiondf-1o 6395 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1o = suc ∅
 
Definitiondf-2o 6396 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2o = suc 1o
 
Definitiondf-3o 6397 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3o = suc 2o
 
Definitiondf-4o 6398 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4o = suc 3o
 
Definitiondf-oadd 6399* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
+o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
 
Definitiondf-omul 6400* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
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