Theorem bnj1500

Description: Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1500.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1500.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1500.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1500.4	⊢ 𝐹 = ∪ 𝐶
Assertion	bnj1500	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1500.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1500.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1500.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1500.4	⊢ 𝐹 = ∪ 𝐶
5		biid	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) )
6		biid	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) )
7		biid	⊢ ( ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) ↔ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) )
8	1 2 3 4 5 6 7	bnj1501	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )