Metamath Proof Explorer


Theorem bnj1312

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 bnj1312.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1312.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1312.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1312.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1312.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1312.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1312.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1312.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
bnj1312.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
bnj1312.10 𝑃 = 𝐻
bnj1312.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1312.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
bnj1312.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
bnj1312.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion bnj1312 ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴𝑓𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 bnj1312.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1312.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1312.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1312.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1312.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1312.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1312.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1312.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 bnj1312.9 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10 bnj1312.10 𝑃 = 𝐻
11 bnj1312.11 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12 bnj1312.12 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺𝑍 ) ⟩ } )
13 bnj1312.13 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14 bnj1312.14 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15 6 simplbi ( 𝜓𝑅 FrSe 𝐴 )
16 5 ssrab3 𝐷𝐴
17 16 a1i ( 𝜓𝐷𝐴 )
18 6 simprbi ( 𝜓𝐷 ≠ ∅ )
19 5 bnj1230 ( 𝑤𝐷 → ∀ 𝑥 𝑤𝐷 )
20 19 bnj1228 ( ( 𝑅 FrSe 𝐴𝐷𝐴𝐷 ≠ ∅ ) → ∃ 𝑥𝐷𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
21 15 17 18 20 syl3anc ( 𝜓 → ∃ 𝑥𝐷𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
22 nfv 𝑥 𝑅 FrSe 𝐴
23 19 nfcii 𝑥 𝐷
24 nfcv 𝑥
25 23 24 nfne 𝑥 𝐷 ≠ ∅
26 22 25 nfan 𝑥 ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ )
27 6 26 nfxfr 𝑥 𝜓
28 27 nf5ri ( 𝜓 → ∀ 𝑥 𝜓 )
29 21 7 28 bnj1521 ( 𝜓 → ∃ 𝑥 𝜒 )
30 7 simp2bi ( 𝜒𝑥𝐷 )
31 5 bnj1538 ( 𝑥𝐷 → ¬ ∃ 𝑓 𝜏 )
32 1 2 3 4 5 6 7 8 9 10 11 12 bnj1489 ( 𝜒𝑄 ∈ V )
33 7 15 bnj835 ( 𝜒𝑅 FrSe 𝐴 )
34 1 2 3 4 5 6 7 8 9 10 bnj1384 ( 𝑅 FrSe 𝐴 → Fun 𝑃 )
35 33 34 syl ( 𝜒 → Fun 𝑃 )
36 1 2 3 4 5 6 7 8 9 10 bnj1415 ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) )
37 35 36 bnj1422 ( 𝜒𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 36 bnj1416 ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
39 1 2 3 4 5 6 7 8 9 10 11 12 35 38 36 bnj1421 ( 𝜒 → Fun 𝑄 )
40 39 38 bnj1422 ( 𝜒𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 37 40 bnj1423 ( 𝜒 → ∀ 𝑧𝐸 ( 𝑄𝑧 ) = ( 𝐺𝑊 ) )
42 14 fneq2i ( 𝑄 Fn 𝐸𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
43 40 42 sylibr ( 𝜒𝑄 Fn 𝐸 )
44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 bnj1452 ( 𝜒𝐸𝐵 )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44 bnj1463 ( 𝜒𝑄𝐶 )
46 45 38 jca ( 𝜒 → ( 𝑄𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
47 1 2 3 4 5 6 7 8 9 10 11 12 46 bnj1491 ( ( 𝜒𝑄 ∈ V ) → ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
48 32 47 mpdan ( 𝜒 → ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
49 48 4 bnj1198 ( 𝜒 → ∃ 𝑓 𝜏 )
50 31 49 nsyl3 ( 𝜒 → ¬ 𝑥𝐷 )
51 29 30 50 bnj1304 ¬ 𝜓
52 6 51 bnj1541 ( 𝑅 FrSe 𝐴𝐷 = ∅ )
53 5 52 bnj1476 ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴𝑓 𝜏 )
54 4 exbii ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
55 df-rex ( ∃ 𝑓𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∃ 𝑓 ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
56 54 55 bitr4i ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
57 56 ralbii ( ∀ 𝑥𝐴𝑓 𝜏 ↔ ∀ 𝑥𝐴𝑓𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
58 53 57 sylib ( 𝑅 FrSe 𝐴 → ∀ 𝑥𝐴𝑓𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )