Metamath Proof Explorer

Theorem bnj207

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 bnj207.1 ⊢ ( 𝜒 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
bnj207.2 ⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 )
bnj207.3 ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 )
bnj207.4 ⊢ ( 𝜒′ ↔ [ 𝑀 / 𝑛 ] 𝜒 )
bnj207.5 ⊢ 𝑀 ∈ V
Assertion bnj207 ( 𝜒′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) ) )

Proof

Step Hyp Ref Expression
1 bnj207.1 ⊢ ( 𝜒 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
2 bnj207.2 ⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 )
3 bnj207.3 ⊢ ( 𝜓′ ↔ [ 𝑀 / 𝑛 ] 𝜓 )
4 bnj207.4 ⊢ ( 𝜒′ ↔ [ 𝑀 / 𝑛 ] 𝜒 )
5 bnj207.5 ⊢ 𝑀 ∈ V
6 1 sbcbii ⊢ ( [ 𝑀 / 𝑛 ] 𝜒 ↔ [ 𝑀 / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
7 nfv ⊢ Ⅎ 𝑛 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 )
8 7 sbc19.21g ⊢ ( 𝑀 ∈ V → ( [ 𝑀 / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝑀 / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) )
9 5 8 ax-mp ⊢ ( [ 𝑀 / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝑀 / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
10 5 bnj89 ⊢ ( [ 𝑀 / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃! 𝑓 [ 𝑀 / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
11 5 bnj90 ⊢ ( [ 𝑀 / 𝑛 ] 𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑀 )
12 11 bicomi ⊢ ( 𝑓 Fn 𝑀 ↔ [ 𝑀 / 𝑛 ] 𝑓 Fn 𝑛 )
13 12 2 3 5 bnj206 ⊢ ( [ 𝑀 / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) )
14 13 eubii ⊢ ( ∃! 𝑓 [ 𝑀 / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) )
15 10 14 bitri ⊢ ( [ 𝑀 / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) )
16 15 imbi2i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 𝑀 / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) ) )
17 9 16 bitri ⊢ ( [ 𝑀 / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) ) )
18 6 17 bitri ⊢ ( [ 𝑀 / 𝑛 ] 𝜒 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) ) )
19 4 18 bitri ⊢ ( 𝜒′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′ ) ) )