Metamath Proof Explorer


Theorem bnj1445

Description: Technical lemma for bnj60 . 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 bnj1445.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1445.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1445.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1445.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1445.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1445.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1445.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1445.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
bnj1445.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
bnj1445.10 𝑃 = 𝐻
bnj1445.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1445.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
bnj1445.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
bnj1445.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
bnj1445.15 ( 𝜒𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
bnj1445.16 ( 𝜒𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
bnj1445.17 ( 𝜃 ↔ ( 𝜒𝑧𝐸 ) )
bnj1445.18 ( 𝜂 ↔ ( 𝜃𝑧 ∈ { 𝑥 } ) )
bnj1445.19 ( 𝜁 ↔ ( 𝜃𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
bnj1445.20 ( 𝜌 ↔ ( 𝜁𝑓𝐻𝑧 ∈ dom 𝑓 ) )
bnj1445.21 ( 𝜎 ↔ ( 𝜌𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
bnj1445.22 ( 𝜑 ↔ ( 𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) )
bnj1445.23 𝑋 = ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
Assertion bnj1445 ( 𝜎 → ∀ 𝑑 𝜎 )

Proof

Step Hyp Ref Expression
1 bnj1445.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1445.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1445.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1445.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1445.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1445.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1445.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1445.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 bnj1445.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10 bnj1445.10 𝑃 = 𝐻
11 bnj1445.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12 bnj1445.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
13 bnj1445.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14 bnj1445.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15 bnj1445.15 ( 𝜒𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16 bnj1445.16 ( 𝜒𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17 bnj1445.17 ( 𝜃 ↔ ( 𝜒𝑧𝐸 ) )
18 bnj1445.18 ( 𝜂 ↔ ( 𝜃𝑧 ∈ { 𝑥 } ) )
19 bnj1445.19 ( 𝜁 ↔ ( 𝜃𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
20 bnj1445.20 ( 𝜌 ↔ ( 𝜁𝑓𝐻𝑧 ∈ dom 𝑓 ) )
21 bnj1445.21 ( 𝜎 ↔ ( 𝜌𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
22 bnj1445.22 ( 𝜑 ↔ ( 𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) )
23 bnj1445.23 𝑋 = ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
24 nfv 𝑑 𝑅 FrSe 𝐴
25 nfre1 𝑑𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) )
26 25 nfab 𝑑 { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
27 3 26 nfcxfr 𝑑 𝐶
28 27 nfcri 𝑑 𝑓𝐶
29 nfv 𝑑 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
30 28 29 nfan 𝑑 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
31 4 30 nfxfr 𝑑 𝜏
32 31 nfex 𝑑𝑓 𝜏
33 32 nfn 𝑑 ¬ ∃ 𝑓 𝜏
34 nfcv 𝑑 𝐴
35 33 34 nfrabw 𝑑 { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
36 5 35 nfcxfr 𝑑 𝐷
37 nfcv 𝑑
38 36 37 nfne 𝑑 𝐷 ≠ ∅
39 24 38 nfan 𝑑 ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ )
40 6 39 nfxfr 𝑑 𝜓
41 36 nfcri 𝑑 𝑥𝐷
42 nfv 𝑑 ¬ 𝑦 𝑅 𝑥
43 36 42 nfralw 𝑑𝑦𝐷 ¬ 𝑦 𝑅 𝑥
44 40 41 43 nf3an 𝑑 ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
45 7 44 nfxfr 𝑑 𝜒
46 45 nf5ri ( 𝜒 → ∀ 𝑑 𝜒 )
47 46 bnj1351 ( ( 𝜒𝑧𝐸 ) → ∀ 𝑑 ( 𝜒𝑧𝐸 ) )
48 47 nf5i 𝑑 ( 𝜒𝑧𝐸 )
49 17 48 nfxfr 𝑑 𝜃
50 nfv 𝑑 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 )
51 49 50 nfan 𝑑 ( 𝜃𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
52 19 51 nfxfr 𝑑 𝜁
53 nfcv 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 )
54 nfcv 𝑑 𝑦
55 54 31 nfsbcw 𝑑 [ 𝑦 / 𝑥 ] 𝜏
56 8 55 nfxfr 𝑑 𝜏′
57 53 56 nfrex 𝑑𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
58 57 nfab 𝑑 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
59 9 58 nfcxfr 𝑑 𝐻
60 59 nfcri 𝑑 𝑓𝐻
61 nfv 𝑑 𝑧 ∈ dom 𝑓
62 52 60 61 nf3an 𝑑 ( 𝜁𝑓𝐻𝑧 ∈ dom 𝑓 )
63 20 62 nfxfr 𝑑 𝜌
64 63 nf5ri ( 𝜌 → ∀ 𝑑 𝜌 )
65 ax-5 ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∀ 𝑑 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) )
66 28 nf5ri ( 𝑓𝐶 → ∀ 𝑑 𝑓𝐶 )
67 ax-5 ( dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) → ∀ 𝑑 dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
68 64 65 66 67 bnj982 ( ( 𝜌𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∀ 𝑑 ( 𝜌𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
69 21 68 hbxfrbi ( 𝜎 → ∀ 𝑑 𝜎 )