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Metamath Proof Explorer

Theorem bnj543

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 bnj543.1 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
bnj543.2 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj543.3 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
bnj543.4 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
bnj543.5 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
Assertion bnj543 ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )

Proof

Step Hyp Ref Expression
1 bnj543.1 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2 bnj543.2 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj543.3 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
4 bnj543.4 ⊢ ( 𝜏 ↔ ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) )
5 bnj543.5 ⊢ ( 𝜎 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) )
6 bnj257 ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑓 Fn 𝑚 ∧ 𝑛 = suc 𝑚 ) )
7 bnj268 ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑓 Fn 𝑚 ∧ 𝑛 = suc 𝑚 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) )
8 6 7 bitri ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) )
9 bnj253 ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) )
10 bnj256 ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) ↔ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ) ∧ ( ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) ) )
11 8 9 10 3bitr3i ⊢ ( ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ) ∧ ( ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) ) )
12 bnj256 ⊢ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ) )
13 12 3anbi1i ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) )
14 bnj170 ⊢ ( ( 𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ) )
15 4 14 bitri ⊢ ( 𝜏 ↔ ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ) )
16 3anan32 ⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚 ) ↔ ( ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) )
17 5 16 bitri ⊢ ( 𝜎 ↔ ( ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) )
18 15 17 anbi12i ⊢ ( ( 𝜏 ∧ 𝜎 ) ↔ ( ( ( 𝜑′ ∧ 𝜓′ ) ∧ 𝑓 Fn 𝑚 ) ∧ ( ( 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ) ) )
19 11 13 18 3bitr4ri ⊢ ( ( 𝜏 ∧ 𝜎 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) )
20 19 anbi2i ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜏 ∧ 𝜎 ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ) )
21 3anass ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( 𝜏 ∧ 𝜎 ) ) )
22 bnj252 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ) )
23 20 21 22 3bitr4i ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) )
24 df-suc ⊢ suc 𝑚 = ( 𝑚 ∪ { 𝑚 } )
25 24 eqeq2i ⊢ ( 𝑛 = suc 𝑚 ↔ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) )
26 25 3anbi2i ⊢ ( ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) )
27 26 anbi2i ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) ) )
28 bnj252 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) ) )
29 27 22 28 3bitr4i ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) ↔ ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) )
30 biid ⊢ ( ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) )
31 1 2 3 30 bnj535 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) → 𝐺 Fn 𝑛 )
32 29 31 sylbi ⊢ ( ( 𝑅 FrSe 𝐴 ∧ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚 ) → 𝐺 Fn 𝑛 )
33 23 32 sylbi ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎 ) → 𝐺 Fn 𝑛 )