Metamath Proof Explorer


Theorem bnj1021

Description: Technical lemma for bnj69 . 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 bnj1021.1 ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
bnj1021.2 ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖𝑛 → ( 𝑓 ‘ suc 𝑖 ) = 𝑦 ∈ ( 𝑓𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj1021.3 ( 𝜒 ↔ ( 𝑛𝐷𝑓 Fn 𝑛𝜑𝜓 ) )
bnj1021.4 ( 𝜃 ↔ ( 𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj1021.5 ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛 ) )
bnj1021.6 ( 𝜂 ↔ ( 𝑖𝑛𝑦 ∈ ( 𝑓𝑖 ) ) )
bnj1021.13 𝐷 = ( ω ∖ { ∅ } )
bnj1021.14 𝐵 = { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) }
Assertion bnj1021 𝑓𝑛𝑖𝑚 ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) )

Proof

Step Hyp Ref Expression
1 bnj1021.1 ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2 bnj1021.2 ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖𝑛 → ( 𝑓 ‘ suc 𝑖 ) = 𝑦 ∈ ( 𝑓𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj1021.3 ( 𝜒 ↔ ( 𝑛𝐷𝑓 Fn 𝑛𝜑𝜓 ) )
4 bnj1021.4 ( 𝜃 ↔ ( 𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5 bnj1021.5 ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛 ) )
6 bnj1021.6 ( 𝜂 ↔ ( 𝑖𝑛𝑦 ∈ ( 𝑓𝑖 ) ) )
7 bnj1021.13 𝐷 = ( ω ∖ { ∅ } )
8 bnj1021.14 𝐵 = { 𝑓 ∣ ∃ 𝑛𝐷 ( 𝑓 Fn 𝑛𝜑𝜓 ) }
9 1 2 3 4 5 6 7 8 bnj996 𝑓𝑛𝑖𝑚𝑝 ( 𝜃 → ( 𝜒𝜏𝜂 ) )
10 anclb ( ( 𝜃 → ( 𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒𝜏𝜂 ) ) ) )
11 bnj252 ( ( 𝜃𝜒𝜏𝜂 ) ↔ ( 𝜃 ∧ ( 𝜒𝜏𝜂 ) ) )
12 11 imbi2i ( ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒𝜏𝜂 ) ) ) )
13 10 12 bitr4i ( ( 𝜃 → ( 𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) )
14 13 2exbii ( ∃ 𝑚𝑝 ( 𝜃 → ( 𝜒𝜏𝜂 ) ) ↔ ∃ 𝑚𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) )
15 14 3exbii ( ∃ 𝑓𝑛𝑖𝑚𝑝 ( 𝜃 → ( 𝜒𝜏𝜂 ) ) ↔ ∃ 𝑓𝑛𝑖𝑚𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) )
16 9 15 mpbi 𝑓𝑛𝑖𝑚𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) )
17 19.37v ( ∃ 𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑝 ( 𝜃𝜒𝜏𝜂 ) ) )
18 bnj1019 ( ∃ 𝑝 ( 𝜃𝜒𝜏𝜂 ) ↔ ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) )
19 18 imbi2i ( ( 𝜃 → ∃ 𝑝 ( 𝜃𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
20 17 19 bitri ( ∃ 𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) ↔ ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
21 20 2exbii ( ∃ 𝑖𝑚𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) ↔ ∃ 𝑖𝑚 ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
22 21 2exbii ( ∃ 𝑓𝑛𝑖𝑚𝑝 ( 𝜃 → ( 𝜃𝜒𝜏𝜂 ) ) ↔ ∃ 𝑓𝑛𝑖𝑚 ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
23 16 22 mpbi 𝑓𝑛𝑖𝑚 ( 𝜃 → ( 𝜃𝜒𝜂 ∧ ∃ 𝑝 𝜏 ) )