This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bnj124

Description: Technical lemma for bnj150 . 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) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)

Ref Expression
Hypotheses bnj124.1 ⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
bnj124.2 ⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
bnj124.3 ⊢ ( 𝜓″ ↔ [ 𝐹 / 𝑓 ] 𝜓′ )
bnj124.4 ⊢ ( 𝜁″ ↔ [ 𝐹 / 𝑓 ] 𝜁′ )
bnj124.5 ⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
Assertion bnj124 ( 𝜁″ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )

Proof

Step Hyp Ref Expression
1 bnj124.1 ⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
2 bnj124.2 ⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
3 bnj124.3 ⊢ ( 𝜓″ ↔ [ 𝐹 / 𝑓 ] 𝜓′ )
4 bnj124.4 ⊢ ( 𝜁″ ↔ [ 𝐹 / 𝑓 ] 𝜁′ )
5 bnj124.5 ⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
6 5 sbcbii ⊢ ( [ 𝐹 / 𝑓 ] 𝜁′ ↔ [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
7 1 bnj95 ⊢ 𝐹 ∈ V
8 nfv ⊢ Ⅎ 𝑓 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 )
9 8 sbc19.21g ⊢ ( 𝐹 ∈ V → ( [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ) )
10 7 9 ax-mp ⊢ ( [ 𝐹 / 𝑓 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )
11 fneq1 ⊢ ( 𝑓 = 𝑧 → ( 𝑓 Fn 1o ↔ 𝑧 Fn 1o ) )
12 fneq1 ⊢ ( 𝑧 = 𝐹 → ( 𝑧 Fn 1o ↔ 𝐹 Fn 1o ) )
13 11 12 sbcie2g ⊢ ( 𝐹 ∈ V → ( [ 𝐹 / 𝑓 ] 𝑓 Fn 1o ↔ 𝐹 Fn 1o ) )
14 7 13 ax-mp ⊢ ( [ 𝐹 / 𝑓 ] 𝑓 Fn 1o ↔ 𝐹 Fn 1o )
15 14 bicomi ⊢ ( 𝐹 Fn 1o ↔ [ 𝐹 / 𝑓 ] 𝑓 Fn 1o )
16 15 2 3 7 bnj206 ⊢ ( [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) )
17 16 imbi2i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝐹 / 𝑓 ] ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )
18 6 10 17 3bitri ⊢ ( [ 𝐹 / 𝑓 ] 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )
19 4 18 bitri ⊢ ( 𝜁″ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″ ) ) )