Metamath Proof Explorer


Theorem bnj1489

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 bnj1489.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1489.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1489.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1489.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1489.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1489.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1489.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1489.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
bnj1489.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
bnj1489.10 𝑃 = 𝐻
bnj1489.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1489.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
Assertion bnj1489 ( 𝜒𝑄 ∈ V )

Proof

Step Hyp Ref Expression
1 bnj1489.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1489.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1489.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1489.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1489.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1489.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1489.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1489.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 bnj1489.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10 bnj1489.10 𝑃 = 𝐻
11 bnj1489.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12 bnj1489.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
13 bnj1364 ( 𝑅 FrSe 𝐴𝑅 Se 𝐴 )
14 df-bnj13 ( 𝑅 Se 𝐴 ↔ ∀ 𝑥𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
15 13 14 sylib ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
16 6 15 bnj832 ( 𝜓 → ∀ 𝑥𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
17 7 16 bnj835 ( 𝜒 → ∀ 𝑥𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
18 5 7 bnj1212 ( 𝜒𝑥𝐴 )
19 17 18 bnj1294 ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
20 nfv 𝑦 𝜓
21 nfv 𝑦 𝑥𝐷
22 nfra1 𝑦𝑦𝐷 ¬ 𝑦 𝑅 𝑥
23 20 21 22 nf3an 𝑦 ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
24 7 23 nfxfr 𝑦 𝜒
25 6 simplbi ( 𝜓𝑅 FrSe 𝐴 )
26 7 25 bnj835 ( 𝜒𝑅 FrSe 𝐴 )
27 26 adantr ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 )
28 1 2 3 4 5 6 7 8 bnj1388 ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )
29 28 r19.21bi ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 𝜏′ )
30 nfv 𝑥 𝑅 FrSe 𝐴
31 nfsbc1v 𝑥 [ 𝑦 / 𝑥 ] 𝜏
32 8 31 nfxfr 𝑥 𝜏′
33 32 nfex 𝑥𝑓 𝜏′
34 30 33 nfan 𝑥 ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ )
35 32 nfeuw 𝑥 ∃! 𝑓 𝜏′
36 34 35 nfim 𝑥 ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ )
37 sneq ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } )
38 bnj1318 ( 𝑥 = 𝑦 → trCl ( 𝑥 , 𝐴 , 𝑅 ) = trCl ( 𝑦 , 𝐴 , 𝑅 ) )
39 37 38 uneq12d ( 𝑥 = 𝑦 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
40 39 eqeq2d ( 𝑥 = 𝑦 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
41 40 anbi2d ( 𝑥 = 𝑦 → ( ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
42 1 2 3 4 8 bnj1373 ( 𝜏′ ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
43 41 42 bitr4di ( 𝑥 = 𝑦 → ( ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ 𝜏′ ) )
44 43 exbidv ( 𝑥 = 𝑦 → ( ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃ 𝑓 𝜏′ ) )
45 44 anbi2d ( 𝑥 = 𝑦 → ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) ) )
46 43 eubidv ( 𝑥 = 𝑦 → ( ∃! 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃! 𝑓 𝜏′ ) )
47 45 46 imbi12d ( 𝑥 = 𝑦 → ( ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ ) ) )
48 biid ( ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
49 1 2 3 48 bnj1321 ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
50 36 47 49 chvarfv ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ )
51 27 29 50 syl2anc ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃! 𝑓 𝜏′ )
52 51 ex ( 𝜒 → ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∃! 𝑓 𝜏′ ) )
53 24 52 ralrimi ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ )
54 9 a1i ( 𝜒𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } )
55 biid ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) ↔ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) )
56 55 bnj1366 ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) → 𝐻 ∈ V )
57 19 53 54 56 syl3anc ( 𝜒𝐻 ∈ V )
58 57 uniexd ( 𝜒 𝐻 ∈ V )
59 10 58 eqeltrid ( 𝜒𝑃 ∈ V )
60 snex { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ∈ V
61 60 a1i ( 𝜒 → { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ∈ V )
62 59 61 bnj1149 ( 𝜒 → ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) ∈ V )
63 12 62 eqeltrid ( 𝜒𝑄 ∈ V )