Theorem wfr3

Description: The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that F is unique. We do this by showing that any function H with the same properties we proved of F in wfr1 and wfr2 is identical to F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr3.1	⊢ 𝑅 We 𝐴
	wfr3.2	⊢ 𝑅 Se 𝐴
	wfr3.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion	wfr3	⊢ ( ( 𝐻 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐻 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐻 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → 𝐹 = 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		wfr3.1	⊢ 𝑅 We 𝐴
2		wfr3.2	⊢ 𝑅 Se 𝐴
3		wfr3.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
4	1 2	pm3.2i	⊢ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 )
5	1 2 3	wfr1	⊢ 𝐹 Fn 𝐴
6	1 2 3	wfr2	⊢ ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
7	6	rgen	⊢ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
8	5 7	pm3.2i	⊢ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
9		wfr3g	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) ∧ ( 𝐻 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐻 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐻 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) ) → 𝐹 = 𝐻 )
10	4 8 9	mp3an12	⊢ ( ( 𝐻 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐻 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐻 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → 𝐹 = 𝐻 )