Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1388.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1388.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1388.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1388.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1388.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1388.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1388.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1388.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
nfv |
⊢ Ⅎ 𝑦 𝜓 |
10 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ 𝐷 |
11 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 |
12 |
9 10 11
|
nf3an |
⊢ Ⅎ 𝑦 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
13 |
7 12
|
nfxfr |
⊢ Ⅎ 𝑦 𝜒 |
14 |
|
bnj1152 |
⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) ) |
15 |
14
|
simplbi |
⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → 𝑦 ∈ 𝐴 ) |
16 |
15
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦 ∈ 𝐴 ) |
17 |
14
|
biimpi |
⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) ) |
19 |
18
|
simprd |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦 𝑅 𝑥 ) |
20 |
7
|
simp3bi |
⊢ ( 𝜒 → ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
21 |
20
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
22 |
|
df-ral |
⊢ ( ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) ) |
23 |
|
con2b |
⊢ ( ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) |
24 |
23
|
albii |
⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) |
25 |
22 24
|
bitri |
⊢ ( ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) |
26 |
|
sp |
⊢ ( ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) → ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) |
27 |
26
|
impcom |
⊢ ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) → ¬ 𝑦 ∈ 𝐷 ) |
28 |
25 27
|
sylan2b |
⊢ ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) → ¬ 𝑦 ∈ 𝐷 ) |
29 |
19 21 28
|
syl2anc |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ 𝑦 ∈ 𝐷 ) |
30 |
5
|
eleq2i |
⊢ ( 𝑦 ∈ 𝐷 ↔ 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } ) |
31 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
32 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
33 |
|
nfsbc1v |
⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜏 |
34 |
8 33
|
nfxfr |
⊢ Ⅎ 𝑥 𝜏′ |
35 |
34
|
nfex |
⊢ Ⅎ 𝑥 ∃ 𝑓 𝜏′ |
36 |
35
|
nfn |
⊢ Ⅎ 𝑥 ¬ ∃ 𝑓 𝜏′ |
37 |
|
sbceq1a |
⊢ ( 𝑥 = 𝑦 → ( 𝜏 ↔ [ 𝑦 / 𝑥 ] 𝜏 ) ) |
38 |
37 8
|
bitr4di |
⊢ ( 𝑥 = 𝑦 → ( 𝜏 ↔ 𝜏′ ) ) |
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 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } ↔ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) ) |
42 |
30 41
|
bitri |
⊢ ( 𝑦 ∈ 𝐷 ↔ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) ) |
43 |
29 42
|
sylnib |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) ) |
44 |
|
iman |
⊢ ( ( 𝑦 ∈ 𝐴 → ∃ 𝑓 𝜏′ ) ↔ ¬ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) ) |
45 |
43 44
|
sylibr |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦 ∈ 𝐴 → ∃ 𝑓 𝜏′ ) ) |
46 |
16 45
|
mpd |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ 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′ ) |