Metamath Proof Explorer

Theorem bnj540

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 bnj540.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj540.2 ⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓 )
bnj540.3 ⊢ 𝐺 ∈ V
Assertion bnj540 ( 𝜓″ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 bnj540.1 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2 bnj540.2 ⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓 )
3 bnj540.3 ⊢ 𝐺 ∈ V
4 1 sbcbii ⊢ ( [ 𝐺 / 𝑓 ] 𝜓 ↔ [ 𝐺 / 𝑓 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5 3 bnj538 ⊢ ( [ 𝐺 / 𝑓 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω [ 𝐺 / 𝑓 ] ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
6 sbcimg ⊢ ( 𝐺 ∈ V → ( [ 𝐺 / 𝑓 ] ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
7 3 6 ax-mp ⊢ ( [ 𝐺 / 𝑓 ] ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
8 7 ralbii ⊢ ( ∀ 𝑖 ∈ ω [ 𝐺 / 𝑓 ] ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
9 4 5 8 3bitri ⊢ ( [ 𝐺 / 𝑓 ] 𝜓 ↔ ∀ 𝑖 ∈ ω ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
10 3 bnj525 ⊢ ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 ↔ suc 𝑖 ∈ 𝑁 )
11 fveq1 ⊢ ( 𝑓 = 𝐺 → ( 𝑓 ‘ suc 𝑖 ) = ( 𝐺 ‘ suc 𝑖 ) )
12 fveq1 ⊢ ( 𝑓 = 𝐺 → ( 𝑓 ‘ 𝑖 ) = ( 𝐺 ‘ 𝑖 ) )
13 12 bnj1113 ⊢ ( 𝑓 = 𝐺 → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
14 11 13 eqeq12d ⊢ ( 𝑓 = 𝐺 → ( ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
15 3 14 sbcie ⊢ ( [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
16 10 15 imbi12i ⊢ ( ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
17 16 ralbii ⊢ ( ∀ 𝑖 ∈ ω ( [ 𝐺 / 𝑓 ] suc 𝑖 ∈ 𝑁 → [ 𝐺 / 𝑓 ] ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
18 2 9 17 3bitri ⊢ ( 𝜓″ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )