Theorem wfr3OLD

Description: Obsolete form of wfr3 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr3OLD.1	⊢ 𝑅 We 𝐴
	wfr3OLD.2	⊢ 𝑅 Se 𝐴
	wfr3OLD.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion	wfr3OLD	⊢ ( ( 𝐻 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐻 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐻 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → 𝐹 = 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		wfr3OLD.1	⊢ 𝑅 We 𝐴
2		wfr3OLD.2	⊢ 𝑅 Se 𝐴
3		wfr3OLD.3	⊢ 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
4	1 2	pm3.2i	⊢ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 )
5	3	wfr1	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )
6	1 2 5	mp2an	⊢ 𝐹 Fn 𝐴
7	3	wfr2	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
8	1 2 7	mpanl12	⊢ ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
9	8	rgen	⊢ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
10	6 9	pm3.2i	⊢ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
11		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 )
12	4 10 11	mp3an12	⊢ ( ( 𝐻 Fn 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐻 ‘ 𝑧 ) = ( 𝐺 ‘ ( 𝐻 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → 𝐹 = 𝐻 )