Metamath Proof Explorer


Theorem bnj1388

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 bnj1388.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
bnj1388.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
bnj1388.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
bnj1388.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
bnj1388.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
bnj1388.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
bnj1388.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
bnj1388.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
Assertion bnj1388 ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )

Proof

Step Hyp Ref Expression
1 bnj1388.1 𝐵 = { 𝑑 ∣ ( 𝑑𝐴 ∧ ∀ 𝑥𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2 bnj1388.2 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3 bnj1388.3 𝐶 = { 𝑓 ∣ ∃ 𝑑𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥𝑑 ( 𝑓𝑥 ) = ( 𝐺𝑌 ) ) }
4 bnj1388.4 ( 𝜏 ↔ ( 𝑓𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5 bnj1388.5 𝐷 = { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6 bnj1388.6 ( 𝜓 ↔ ( 𝑅 FrSe 𝐴𝐷 ≠ ∅ ) )
7 bnj1388.7 ( 𝜒 ↔ ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8 bnj1388.8 ( 𝜏′[ 𝑦 / 𝑥 ] 𝜏 )
9 nfv 𝑦 𝜓
10 nfv 𝑦 𝑥𝐷
11 nfra1 𝑦𝑦𝐷 ¬ 𝑦 𝑅 𝑥
12 9 10 11 nf3an 𝑦 ( 𝜓𝑥𝐷 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
13 7 12 nfxfr 𝑦 𝜒
14 bnj1152 ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑦𝐴𝑦 𝑅 𝑥 ) )
15 14 simplbi ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → 𝑦𝐴 )
16 15 adantl ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦𝐴 )
17 14 biimpi ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ( 𝑦𝐴𝑦 𝑅 𝑥 ) )
18 17 adantl ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦𝐴𝑦 𝑅 𝑥 ) )
19 18 simprd ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦 𝑅 𝑥 )
20 7 simp3bi ( 𝜒 → ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
21 20 adantr ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 )
22 df-ral ( ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦𝐷 → ¬ 𝑦 𝑅 𝑥 ) )
23 con2b ( ( 𝑦𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) )
24 23 albii ( ∀ 𝑦 ( 𝑦𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) )
25 22 24 bitri ( ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) )
26 sp ( ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) → ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) )
27 26 impcom ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦𝐷 ) ) → ¬ 𝑦𝐷 )
28 25 27 sylan2b ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦𝐷 ¬ 𝑦 𝑅 𝑥 ) → ¬ 𝑦𝐷 )
29 19 21 28 syl2anc ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ 𝑦𝐷 )
30 5 eleq2i ( 𝑦𝐷𝑦 ∈ { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 } )
31 nfcv 𝑥 𝑦
32 nfcv 𝑥 𝐴
33 nfsbc1v 𝑥 [ 𝑦 / 𝑥 ] 𝜏
34 8 33 nfxfr 𝑥 𝜏′
35 34 nfex 𝑥𝑓 𝜏′
36 35 nfn 𝑥 ¬ ∃ 𝑓 𝜏′
37 sbceq1a ( 𝑥 = 𝑦 → ( 𝜏[ 𝑦 / 𝑥 ] 𝜏 ) )
38 37 8 bitr4di ( 𝑥 = 𝑦 → ( 𝜏𝜏′ ) )
39 38 exbidv ( 𝑥 = 𝑦 → ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 𝜏′ ) )
40 39 notbid ( 𝑥 = 𝑦 → ( ¬ ∃ 𝑓 𝜏 ↔ ¬ ∃ 𝑓 𝜏′ ) )
41 31 32 36 40 elrabf ( 𝑦 ∈ { 𝑥𝐴 ∣ ¬ ∃ 𝑓 𝜏 } ↔ ( 𝑦𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
42 30 41 bitri ( 𝑦𝐷 ↔ ( 𝑦𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
43 29 42 sylnib ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ ( 𝑦𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
44 iman ( ( 𝑦𝐴 → ∃ 𝑓 𝜏′ ) ↔ ¬ ( 𝑦𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
45 43 44 sylibr ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦𝐴 → ∃ 𝑓 𝜏′ ) )
46 16 45 mpd ( ( 𝜒𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 𝜏′ )
47 46 ex ( 𝜒 → ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∃ 𝑓 𝜏′ ) )
48 13 47 ralrimi ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )