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 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1312.2 ⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1312.3 ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
bnj1312.4 ⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1312.5 ⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1312.6 ⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
bnj1312.7 ⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 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 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
bnj1312.13 ⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
bnj1312.14 ⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion bnj1312 ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 bnj1312.1 ⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1312.2 ⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1312.3 ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4 bnj1312.4 ⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1312.5 ⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1312.6 ⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7 bnj1312.7 ⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 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 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13 bnj1312.13 ⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14 bnj1312.14 ⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15 6 simplbi ⊢ ( 𝜓 → 𝑅 FrSe 𝐴 )
16 5 ssrab3 ⊢ 𝐷 ⊆ 𝐴
17 16 a1i ⊢ ( 𝜓 → 𝐷 ⊆ 𝐴 )
18 6 simprbi ⊢ ( 𝜓 → 𝐷 ≠ ∅ )
19 5 bnj1230 ⊢ ( 𝑤 ∈ 𝐷 → ∀ 𝑥 𝑤 ∈ 𝐷 )
20 19 bnj1228 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
21 15 17 18 20 syl3anc ⊢ ( 𝜓 → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
22 nfv ⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴
23 19 nfcii ⊢ Ⅎ 𝑥 𝐷
24 nfcv ⊢ Ⅎ 𝑥 ∅
25 23 24 nfne ⊢ Ⅎ 𝑥 𝐷 ≠ ∅
26 22 25 nfan ⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ )
27 6 26 nfxfr ⊢ Ⅎ 𝑥 𝜓
28 27 nf5ri ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
29 21 7 28 bnj1521 ⊢ ( 𝜓 → ∃ 𝑥 𝜒 )
30 7 simp2bi ⊢ ( 𝜒 → 𝑥 ∈ 𝐷 )
31 5 bnj1538 ⊢ ( 𝑥 ∈ 𝐷 → ¬ ∃ 𝑓 𝜏 )
32 1 2 3 4 5 6 7 8 9 10 11 12 bnj1489 ⊢ ( 𝜒 → 𝑄 ∈ V )
33 7 15 bnj835 ⊢ ( 𝜒 → 𝑅 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 ⊢ ( 𝜒 → 𝑃 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 ⊢ ( 𝜒 → 𝑄 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 → ∀ 𝑧 ∈ 𝐸 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )
42 14 fneq2i ⊢ ( 𝑄 Fn 𝐸 ↔ 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
43 40 42 sylibr ⊢ ( 𝜒 → 𝑄 Fn 𝐸 )
44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 bnj1452 ⊢ ( 𝜒 → 𝐸 ∈ 𝐵 )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44 bnj1463 ⊢ ( 𝜒 → 𝑄 ∈ 𝐶 )
46 45 38 jca ⊢ ( 𝜒 → ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
47 1 2 3 4 5 6 7 8 9 10 11 12 46 bnj1491 ⊢ ( ( 𝜒 ∧ 𝑄 ∈ V ) → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
48 32 47 mpdan ⊢ ( 𝜒 → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
49 48 4 bnj1198 ⊢ ( 𝜒 → ∃ 𝑓 𝜏 )
50 31 49 nsyl3 ⊢ ( 𝜒 → ¬ 𝑥 ∈ 𝐷 )
51 29 30 50 bnj1304 ⊢ ¬ 𝜓
52 6 51 bnj1541 ⊢ ( 𝑅 FrSe 𝐴 → 𝐷 = ∅ )
53 5 52 bnj1476 ⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 )
54 4 exbii ⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
55 df-rex ⊢ ( ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
56 54 55 bitr4i ⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
57 56 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
58 53 57 sylib ⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )