bnj969

Description: Technical lemma for bnj69 . 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	bnj969.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj969.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj969.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj969.10	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj969.12	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
	bnj969.14	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj969.15	⊢ ( 𝜎 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛 ) )
Assertion	bnj969	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝐶 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		bnj969.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj969.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj969.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj969.10	⊢ 𝐷 = ( ω ∖ { ∅ } )
5		bnj969.12	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
6		bnj969.14	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
7		bnj969.15	⊢ ( 𝜎 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛 ) )
8		simpl	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
9		bnj667	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
10	9 3 6	3imtr4i	⊢ ( 𝜒 → 𝜏 )
11	10	3ad2ant1	⊢ ( ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) → 𝜏 )
12	11	adantl	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝜏 )
13	3	bnj1232	⊢ ( 𝜒 → 𝑛 ∈ 𝐷 )
14		vex	⊢ 𝑚 ∈ V
15	14	bnj216	⊢ ( 𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛 )
16		id	⊢ ( 𝑝 = suc 𝑛 → 𝑝 = suc 𝑛 )
17	13 15 16	3anim123i	⊢ ( ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) → ( 𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛 ) )
18		3ancomb	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛 ) ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛 ) )
19	7 18	bitri	⊢ ( 𝜎 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛 ) )
20	17 19	sylibr	⊢ ( ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) → 𝜎 )
21	20	adantl	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝜎 )
22	8 12 21	jca32	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜏 ∧ 𝜎 ) ) )
23		bnj256	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜏 ∧ 𝜎 ) ) )
24	22 23	sylibr	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) )
25	4 6 7 1 2	bnj938	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
26	5 25	eqeltrid	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐶 ∈ V )
27	24 26	syl	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝐶 ∈ V )

Metamath Proof Explorer

Theorem bnj969

Proof