Theorem bnj544

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj544.1	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj544.2	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj544.3	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj544.4	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
	bnj544.5	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
	bnj544.6	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
Assertion	bnj544	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )

Proof

Step	Hyp	Ref	Expression
1		bnj544.1	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2		bnj544.2	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj544.3	⊢ 𝐷 = ( ω ∖ { ∅ } )
4		bnj544.4	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
5		bnj544.5	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
6		bnj544.6	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
7	3	bnj923	⊢ ( 𝑚 ∈ 𝐷 → 𝑚 ∈ ω )
8	7	3anim1i	⊢ ( ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) → ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
9	6 8	sylbi	⊢ ( 𝜎 → ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
10		biid	⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
11	1 2 4 5 10	bnj543	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) ) → 𝐺 Fn 𝑛 )
12	9 11	syl3an3	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )