Metamath Proof Explorer

Theorem bnj983

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 bnj983.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
bnj983.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj983.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
bnj983.4 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
bnj983.5 ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
Assertion bnj983 ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )

Proof

Step Hyp Ref Expression
1 bnj983.1 ⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2 bnj983.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj983.3 ⊢ 𝐷 = ( ω ∖ { ∅ } )
4 bnj983.4 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
5 bnj983.5 ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
6 1 2 3 4 bnj882 ⊢ trCl ( 𝑋 , 𝐴 , 𝑅 ) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 )
7 6 eleq2i ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ 𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
8 eliun ⊢ ( 𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) )
9 eliun ⊢ ( 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) )
10 9 rexbii ⊢ ( ∃ 𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑓 ∈ 𝐵 ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) )
11 8 10 bitri ⊢ ( 𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓 ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑓 ∈ 𝐵 ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) )
12 df-rex ⊢ ( ∃ 𝑓 ∈ 𝐵 ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐵 ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
13 4 abeq2i ⊢ ( 𝑓 ∈ 𝐵 ↔ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
14 13 anbi1i ⊢ ( ( 𝑓 ∈ 𝐵 ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
15 14 exbii ⊢ ( ∃ 𝑓 ( 𝑓 ∈ 𝐵 ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ∃ 𝑓 ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
16 12 15 bitri ⊢ ( ∃ 𝑓 ∈ 𝐵 ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑓 ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
17 7 11 16 3bitri ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
18 bnj252 ⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
19 5 18 bitri ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
20 19 exbii ⊢ ( ∃ 𝑛 𝜒 ↔ ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
21 20 anbi1i ⊢ ( ( ∃ 𝑛 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
22 df-rex ⊢ ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
23 df-rex ⊢ ( ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ↔ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
24 22 23 anbi12i ⊢ ( ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
25 21 24 bitr4i ⊢ ( ( ∃ 𝑛 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑖 ∈ dom 𝑓 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
26 17 25 bnj133 ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ( ∃ 𝑛 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
27 19.41v ⊢ ( ∃ 𝑛 ( 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ∃ 𝑛 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
28 26 27 bnj133 ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ∃ 𝑛 ( 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
29 2 bnj1095 ⊢ ( 𝜓 → ∀ 𝑖 𝜓 )
30 29 5 bnj1096 ⊢ ( 𝜒 → ∀ 𝑖 𝜒 )
31 30 nf5i ⊢ Ⅎ 𝑖 𝜒
32 31 19.42 ⊢ ( ∃ 𝑖 ( 𝜒 ∧ ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
33 32 2exbii ⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑓 ∃ 𝑛 ( 𝜒 ∧ ∃ 𝑖 ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
34 28 33 bitr4i ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
35 3anass ⊢ ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
36 35 3exbii ⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ ( 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
37 fndm ⊢ ( 𝑓 Fn 𝑛 → dom 𝑓 = 𝑛 )
38 5 37 bnj770 ⊢ ( 𝜒 → dom 𝑓 = 𝑛 )
39 eleq2 ⊢ ( dom 𝑓 = 𝑛 → ( 𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛 ) )
40 39 3anbi2d ⊢ ( dom 𝑓 = 𝑛 → ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
41 38 40 syl ⊢ ( 𝜒 → ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
42 41 3ad2ant1 ⊢ ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) → ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
43 42 ibi ⊢ ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) → ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
44 41 3ad2ant1 ⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) → ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
45 44 ibir ⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) → ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
46 43 45 impbii ⊢ ( ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
47 46 3exbii ⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )
48 34 36 47 3bitr2i ⊢ ( 𝑍 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ ( 𝑓 ‘ 𝑖 ) ) )