Metamath Proof Explorer


Theorem bnj1421

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 bnj1421.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1421.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1421.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1421.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1421.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1421.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1421.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1421.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
bnj1421.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
bnj1421.10 𝑃 = 𝐻
bnj1421.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1421.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
bnj1421.13 ( 𝜒 → Fun 𝑃 )
bnj1421.14 ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
bnj1421.15 ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion bnj1421 ( 𝜒 → Fun 𝑄 )

Proof

Step Hyp Ref Expression
1 bnj1421.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1421.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1421.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1421.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1421.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1421.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1421.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1421.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 bnj1421.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10 bnj1421.10 𝑃 = 𝐻
11 bnj1421.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12 bnj1421.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
13 bnj1421.13 ( 𝜒 → Fun 𝑃 )
14 bnj1421.14 ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
15 bnj1421.15 ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16 vex 𝑥 ∈ V
17 fvex ( 𝐺𝑍 ) ∈ V
18 16 17 funsn Fun { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ }
19 13 18 jctir ( 𝜒 → ( Fun 𝑃 ∧ Fun { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) )
20 17 dmsnop dom { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } = { 𝑥 }
21 20 a1i ( 𝜒 → dom { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } = { 𝑥 } )
22 15 21 ineq12d ( 𝜒 → ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) = ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) )
23 6 simplbi ( 𝜓𝑅 FrSe 𝐴 )
24 7 23 bnj835 ( 𝜒𝑅 FrSe 𝐴 )
25 biid ( 𝑅 FrSe 𝐴𝑅 FrSe 𝐴 )
26 biid ( ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
27 biid ( ∀ 𝑧𝐴 ( 𝑧 𝑅 𝑥[ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑧𝐴 ( 𝑧 𝑅 𝑥[ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
28 biid ( ( 𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀ 𝑧𝐴 ( 𝑧 𝑅 𝑥[ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀ 𝑧𝐴 ( 𝑧 𝑅 𝑥[ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
29 eqid ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) ) = ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) )
30 25 26 27 28 29 bnj1417 ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
31 disjsn ( ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
32 31 ralbii ( ∀ 𝑥𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ∀ 𝑥𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
33 30 32 sylibr ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
34 24 33 syl ( 𝜒 → ∀ 𝑥𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
35 5 7 bnj1212 ( 𝜒𝑥𝐴 )
36 34 35 bnj1294 ( 𝜒 → ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ )
37 22 36 eqtrd ( 𝜒 → ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) = ∅ )
38 funun ( ( ( Fun 𝑃 ∧ Fun { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) ∧ ( dom 𝑃 ∩ dom { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) = ∅ ) → Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) )
39 19 37 38 syl2anc ( 𝜒 → Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) )
40 12 funeqi ( Fun 𝑄 ↔ Fun ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } ) )
41 39 40 sylibr ( 𝜒 → Fun 𝑄 )