Metamath Proof Explorer


Theorem bnj1452

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 bnj1452.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1452.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1452.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1452.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1452.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1452.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1452.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1452.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
bnj1452.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
bnj1452.10 𝑃 = 𝐻
bnj1452.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1452.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
bnj1452.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
bnj1452.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion bnj1452 ( 𝜒𝐸𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1452.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1452.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1452.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1452.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1452.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1452.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1452.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1452.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 bnj1452.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10 bnj1452.10 𝑃 = 𝐻
11 bnj1452.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12 bnj1452.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
13 bnj1452.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14 bnj1452.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15 5 7 bnj1212 ( 𝜒𝑥𝐴 )
16 15 snssd ( 𝜒 → { 𝑥 } ⊆ 𝐴 )
17 bnj1147 trCl ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐴
18 17 a1i ( 𝜒 → trCl ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐴 )
19 16 18 unssd ( 𝜒 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ⊆ 𝐴 )
20 14 19 eqsstrid ( 𝜒𝐸𝐴 )
21 elsni ( 𝑧 ∈ { 𝑥 } → 𝑧 = 𝑥 )
22 21 adantl ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → 𝑧 = 𝑥 )
23 bnj602 ( 𝑧 = 𝑥 → pred ( 𝑧 , 𝐴 , 𝑅 ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
24 22 23 syl ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
25 6 simplbi ( 𝜓𝑅 FrSe 𝐴 )
26 7 25 bnj835 ( 𝜒𝑅 FrSe 𝐴 )
27 bnj906 ( ( 𝑅 FrSe 𝐴𝑥𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
28 26 15 27 syl2anc ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
29 28 ad2antrr ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
30 24 29 eqsstrd ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
31 ssun4 ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
32 31 14 sseqtrrdi ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 )
33 30 32 syl ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 )
34 26 ad2antrr ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 )
35 simpr ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
36 17 35 bnj1213 ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑧𝐴 )
37 bnj906 ( ( 𝑅 FrSe 𝐴𝑧𝐴 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑧 , 𝐴 , 𝑅 ) )
38 34 36 37 syl2anc ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑧 , 𝐴 , 𝑅 ) )
39 15 ad2antrr ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑥𝐴 )
40 bnj1125 ( ( 𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → trCl ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
41 34 39 35 40 syl3anc ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → trCl ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
42 38 41 sstrd ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
43 42 32 syl ( ( ( 𝜒𝑧𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 )
44 14 bnj1424 ( 𝑧𝐸 → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
45 44 adantl ( ( 𝜒𝑧𝐸 ) → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
46 33 43 45 mpjaodan ( ( 𝜒𝑧𝐸 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 )
47 46 ralrimiva ( 𝜒 → ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 )
48 snex { 𝑥 } ∈ V
49 48 a1i ( 𝜒 → { 𝑥 } ∈ V )
50 bnj893 ( ( 𝑅 FrSe 𝐴𝑥𝐴 ) → trCl ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
51 26 15 50 syl2anc ( 𝜒 → trCl ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
52 49 51 bnj1149 ( 𝜒 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∈ V )
53 14 52 eqeltrid ( 𝜒𝐸 ∈ V )
54 1 bnj1454 ( 𝐸 ∈ V → ( 𝐸𝐵[ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) )
55 53 54 syl ( 𝜒 → ( 𝐸𝐵[ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) )
56 bnj602 ( 𝑥 = 𝑧 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑧 , 𝐴 , 𝑅 ) )
57 56 sseq1d ( 𝑥 = 𝑧 → ( pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) )
58 57 cbvralvw ( ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 )
59 58 anbi2i ( ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) )
60 59 sbcbii ( [ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ [ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) )
61 55 60 bitrdi ( 𝜒 → ( 𝐸𝐵[ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) )
62 sseq1 ( 𝑑 = 𝐸 → ( 𝑑𝐴𝐸𝐴 ) )
63 sseq2 ( 𝑑 = 𝐸 → ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) )
64 63 raleqbi1dv ( 𝑑 = 𝐸 → ( ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) )
65 62 64 anbi12d ( 𝑑 = 𝐸 → ( ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸𝐴 ∧ ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) )
66 65 sbcieg ( 𝐸 ∈ V → ( [ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸𝐴 ∧ ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) )
67 53 66 syl ( 𝜒 → ( [ 𝐸 / 𝑑 ] ( 𝑑𝐴 ∧ ∀ 𝑧𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸𝐴 ∧ ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) )
68 61 67 bitrd ( 𝜒 → ( 𝐸𝐵 ↔ ( 𝐸𝐴 ∧ ∀ 𝑧𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) )
69 20 47 68 mpbir2and ( 𝜒𝐸𝐵 )