bnj938

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	bnj938.1	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj938.2	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
	bnj938.3	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
	bnj938.4	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj938.5	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj938	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		bnj938.1	⊢ 𝐷 = ( ω ∖ { ∅ } )
2		bnj938.2	⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
3		bnj938.3	⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
4		bnj938.4	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
5		bnj938.5	⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
6		elisset	⊢ ( 𝑋 ∈ 𝐴 → ∃ 𝑥 𝑥 = 𝑋 )
7	6	bnj706	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∃ 𝑥 𝑥 = 𝑋 )
8		bnj291	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ∧ 𝑋 ∈ 𝐴 ) )
9	8	simplbi	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) )
10		bnj602	⊢ ( 𝑥 = 𝑋 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
11	10	eqeq2d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
12	11 4	syl6bbr	⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ 𝜑′ ) )
13	12	3anbi2d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) ) )
14	13 2	syl6bbr	⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ 𝜏 ) )
15	14	3anbi2d	⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ) )
16	9 15	syl5ibr	⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) ) )
17		biid	⊢ ( ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ↔ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) )
18		biid	⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
19	1 17 3 18 5	bnj546	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝜓′ ) ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
20	16 19	syl6	⊢ ( 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V ) )
21	20	exlimiv	⊢ ( ∃ 𝑥 𝑥 = 𝑋 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V ) )
22	7 21	mpcom	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )

Metamath Proof Explorer

Theorem bnj938

Proof