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Metamath Proof Explorer

Theorem bnj222

Description: Technical lemma for bnj229 . 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
Hypothesis bnj222.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion bnj222 ( 𝜓 ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 bnj222.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2 suceq ⊢ ( 𝑖 = 𝑚 → suc 𝑖 = suc 𝑚 )
3 2 eleq1d ⊢ ( 𝑖 = 𝑚 → ( suc 𝑖 ∈ 𝑁 ↔ suc 𝑚 ∈ 𝑁 ) )
4 2 fveq2d ⊢ ( 𝑖 = 𝑚 → ( 𝐹 ‘ suc 𝑖 ) = ( 𝐹 ‘ suc 𝑚 ) )
5 fveq2 ⊢ ( 𝑖 = 𝑚 → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑚 ) )
6 5 bnj1113 ⊢ ( 𝑖 = 𝑚 → ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
7 4 6 eqeq12d ⊢ ( 𝑖 = 𝑚 → ( ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
8 3 7 imbi12d ⊢ ( 𝑖 = 𝑚 → ( ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
9 8 cbvralvw ⊢ ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
10 1 9 bitri ⊢ ( 𝜓 ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )