Theorem fpr1

Description: Law of well-founded recursion over a partial order, part one. Establish the functionality and domain of the recursive function generator. Note that by requiring a partial order we can avoid using the axiom of infinity. (Contributed by Scott Fenton, 11-Sep-2023)

	Ref	Expression
Hypothesis	fprr.1	⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion	fpr1	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		fprr.1	⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
2		eqid	⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
3	2	frrlem1	⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) }
4	3 1	fprlem1	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ∧ ℎ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) → ( ( 𝑏 𝑔 𝑢 ∧ 𝑏 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )
5	3 1 4	frrlem9	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → Fun 𝐹 )
6		eqid	⊢ ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ) = ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } )
7		simp1	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
8		ssidd	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ Pred ( 𝑅 , 𝐴 , 𝑧 ) )
9		fprlem2	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → ∀ 𝑦 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ Pred ( 𝑅 , 𝐴 , 𝑧 ) )
10		setlikespec	⊢ ( ( 𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
11	10	ancoms	⊢ ( ( 𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
12	11	3ad2antl3	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ∈ V )
13		predss	⊢ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴
14	13	a1i	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴 )
15		difssd	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴 )
16		simpr	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ )
17	15 16	jca	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ( ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴 ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) )
18		frpomin2	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴 ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
19	17 18	syldan	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
20	3 1 4 6 7 8 9 12 14 19	frrlem14	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = 𝐴 )
21		df-fn	⊢ ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) )
22	5 20 21	sylanbrc	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )