Metamath Proof Explorer

Theorem bnj130

Description: Technical lemma for bnj151 . 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 bnj130.1 ⊢ ( 𝜃 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
bnj130.2 ⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
bnj130.3 ⊢ ( 𝜓′ ↔ [ 1o / 𝑛 ] 𝜓 )
bnj130.4 ⊢ ( 𝜃′ ↔ [ 1o / 𝑛 ] 𝜃 )
Assertion bnj130 ( 𝜃′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )

Proof

Step Hyp Ref Expression
1 bnj130.1 ⊢ ( 𝜃 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
2 bnj130.2 ⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
3 bnj130.3 ⊢ ( 𝜓′ ↔ [ 1o / 𝑛 ] 𝜓 )
4 bnj130.4 ⊢ ( 𝜃′ ↔ [ 1o / 𝑛 ] 𝜃 )
5 1 sbcbii ⊢ ( [ 1o / 𝑛 ] 𝜃 ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
6 bnj105 ⊢ 1o ∈ V
7 6 bnj90 ⊢ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o )
8 7 bicomi ⊢ ( 𝑓 Fn 1o ↔ [ 1o / 𝑛 ] 𝑓 Fn 𝑛 )
9 8 2 3 3anbi123i ⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) )
10 sbc3an ⊢ ( [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) )
11 9 10 bitr4i ⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
12 11 eubii ⊢ ( ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ∃! 𝑓 [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
13 6 bnj89 ⊢ ( [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃! 𝑓 [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
14 12 13 bitr4i ⊢ ( ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
15 14 imbi2i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
16 nfv ⊢ Ⅎ 𝑛 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 )
17 16 sbc19.21g ⊢ ( 1o ∈ V → ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) )
18 6 17 ax-mp ⊢ ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
19 15 18 bitr4i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
20 5 4 19 3bitr4i ⊢ ( 𝜃′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )