Metamath Proof Explorer

Theorem bnj558

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 bnj558.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
bnj558.16 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
bnj558.17 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
bnj558.18 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
bnj558.19 ⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
bnj558.20 ⊢ ( 𝜁 ↔ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖 ) )
bnj558.21 ⊢ 𝐵 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 )
bnj558.22 ⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 )
bnj558.23 ⊢ 𝐾 = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 )
bnj558.24 ⊢ 𝐿 = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 )
bnj558.25 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , 𝐶 ⟩ } )
bnj558.28 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
bnj558.29 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj558.36 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )
Assertion bnj558 ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝐺 ‘ suc 𝑖 ) = 𝐾 )

Proof

Step Hyp Ref Expression
1 bnj558.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
2 bnj558.16 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
3 bnj558.17 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
4 bnj558.18 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
5 bnj558.19 ⊢ ( 𝜂 ↔ ( 𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝 ) )
6 bnj558.20 ⊢ ( 𝜁 ↔ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖 ) )
7 bnj558.21 ⊢ 𝐵 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 )
8 bnj558.22 ⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 )
9 bnj558.23 ⊢ 𝐾 = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 )
10 bnj558.24 ⊢ 𝐿 = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 )
11 bnj558.25 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , 𝐶 ⟩ } )
12 bnj558.28 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
13 bnj558.29 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
14 bnj558.36 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 bnj557 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝐺 ‘ 𝑚 ) = 𝐿 )
16 bnj422 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) ↔ ( 𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) )
17 bnj253 ⊢ ( ( 𝜂 ∧ 𝜁 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) ↔ ( ( 𝜂 ∧ 𝜁 ) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) )
18 16 17 bitri ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) ↔ ( ( 𝜂 ∧ 𝜁 ) ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) )
19 18 simp1bi ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝜂 ∧ 𝜁 ) )
20 5 6 9 10 9 10 bnj554 ⊢ ( ( 𝜂 ∧ 𝜁 ) → ( ( 𝐺 ‘ 𝑚 ) = 𝐿 ↔ ( 𝐺 ‘ suc 𝑖 ) = 𝐾 ) )
21 19 20 syl ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( ( 𝐺 ‘ 𝑚 ) = 𝐿 ↔ ( 𝐺 ‘ suc 𝑖 ) = 𝐾 ) )
22 15 21 mpbid ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝐺 ‘ suc 𝑖 ) = 𝐾 )