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 𝐴𝑥𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛𝜑𝜓 ) ) )
bnj130.2 ( 𝜑′[ 1o / 𝑛 ] 𝜑 )
bnj130.3 ( 𝜓′[ 1o / 𝑛 ] 𝜓 )
bnj130.4 ( 𝜃′[ 1o / 𝑛 ] 𝜃 )
Assertion bnj130 ( 𝜃′ ↔ ( ( 𝑅 FrSe 𝐴𝑥𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o𝜑′𝜓′ ) ) )

Proof

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