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Metamath Proof Explorer

Theorem bnj1529

Description: Technical lemma for bnj1522 . 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 bnj1529.1 ⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
bnj1529.2 ⊢ ( 𝑤 ∈ 𝐹 → ∀ 𝑥 𝑤 ∈ 𝐹 )
Assertion bnj1529 ( 𝜒 → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )

Proof

Step Hyp Ref Expression
1 bnj1529.1 ⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
2 bnj1529.2 ⊢ ( 𝑤 ∈ 𝐹 → ∀ 𝑥 𝑤 ∈ 𝐹 )
3 nfv ⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
4 2 nfcii ⊢ Ⅎ 𝑥 𝐹
5 nfcv ⊢ Ⅎ 𝑥 𝑦
6 4 5 nffv ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 )
7 nfcv ⊢ Ⅎ 𝑥 𝐺
8 nfcv ⊢ Ⅎ 𝑥 pred ( 𝑦 , 𝐴 , 𝑅 )
9 4 8 nfres ⊢ Ⅎ 𝑥 ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) )
10 5 9 nfop ⊢ Ⅎ 𝑥 ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩
11 7 10 nffv ⊢ Ⅎ 𝑥 ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
12 6 11 nfeq ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
13 fveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
14 id ⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
15 bnj602 ⊢ ( 𝑥 = 𝑦 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑦 , 𝐴 , 𝑅 ) )
16 15 reseq2d ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) = ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
17 14 16 opeq12d ⊢ ( 𝑥 = 𝑦 → ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ )
18 17 fveq2d ⊢ ( 𝑥 = 𝑦 → ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )
19 13 18 eqeq12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
20 3 12 19 cbvralw ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )
21 1 20 sylib ⊢ ( 𝜒 → ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ ⟨ 𝑦 , ( 𝐹 ↾ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ⟩ ) )