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Metamath Proof Explorer

Theorem bnj539

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 bnj539.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj539.2 ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 )
bnj539.3 ⊢ 𝑀 ∈ V
Assertion bnj539 ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 bnj539.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2 bnj539.2 ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 )
3 bnj539.3 ⊢ 𝑀 ∈ V
4 1 sbcbii ⊢ ( [ 𝑀 / 𝑛 ] 𝜓 ↔ [ 𝑀 / 𝑛 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5 3 bnj538 ⊢ ( [ 𝑀 / 𝑛 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω [ 𝑀 / 𝑛 ] ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
6 sbcimg ⊢ ( 𝑀 ∈ V → ( [ 𝑀 / 𝑛 ] ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( [ 𝑀 / 𝑛 ] suc 𝑖 ∈ 𝑛 → [ 𝑀 / 𝑛 ] ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
7 3 6 ax-mp ⊢ ( [ 𝑀 / 𝑛 ] ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( [ 𝑀 / 𝑛 ] suc 𝑖 ∈ 𝑛 → [ 𝑀 / 𝑛 ] ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
8 sbcel2gv ⊢ ( 𝑀 ∈ V → ( [ 𝑀 / 𝑛 ] suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑀 ) )
9 3 8 ax-mp ⊢ ( [ 𝑀 / 𝑛 ] suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑀 )
10 3 bnj525 ⊢ ( [ 𝑀 / 𝑛 ] ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
11 9 10 imbi12i ⊢ ( ( [ 𝑀 / 𝑛 ] suc 𝑖 ∈ 𝑛 → [ 𝑀 / 𝑛 ] ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
12 7 11 bitri ⊢ ( [ 𝑀 / 𝑛 ] ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
13 12 ralbii ⊢ ( ∀ 𝑖 ∈ ω [ 𝑀 / 𝑛 ] ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
14 5 13 bitri ⊢ ( [ 𝑀 / 𝑛 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
15 4 14 bitri ⊢ ( [ 𝑀 / 𝑛 ] 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
16 2 15 bitri ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑀 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )