Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1110.3 |
⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
2 |
|
bnj1110.7 |
⊢ 𝐷 = ( ω ∖ { ∅ } ) |
3 |
|
bnj1110.18 |
⊢ ( 𝜎 ↔ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → 𝜂′ ) ) |
4 |
|
bnj1110.19 |
⊢ ( 𝜑0 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) ) |
5 |
|
bnj1110.26 |
⊢ ( 𝜂′ ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) ) |
6 |
2
|
bnj1098 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) |
7 |
|
bnj219 |
⊢ ( 𝑖 = suc 𝑗 → 𝑗 E 𝑖 ) |
8 |
7
|
adantl |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) → 𝑗 E 𝑖 ) |
9 |
8
|
ancli |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ∧ 𝑗 E 𝑖 ) ) |
10 |
|
df-3an |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ↔ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ∧ 𝑗 E 𝑖 ) ) |
11 |
9 10
|
sylibr |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) |
12 |
6 11
|
bnj1023 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) |
13 |
1
|
bnj1232 |
⊢ ( 𝜒 → 𝑛 ∈ 𝐷 ) |
14 |
13
|
3ad2ant3 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) → 𝑛 ∈ 𝐷 ) |
15 |
4
|
bnj1232 |
⊢ ( 𝜑0 → 𝑖 ∈ 𝑛 ) |
16 |
14 15
|
anim12ci |
⊢ ( ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) → ( 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ) |
17 |
16
|
anim2i |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ) ) |
18 |
|
3anass |
⊢ ( ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ↔ ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ) ) |
19 |
17 18
|
sylibr |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ) |
20 |
12 19
|
bnj1101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) |
21 |
|
3simpb |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) → ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) ) |
22 |
4
|
bnj1235 |
⊢ ( 𝜑0 → 𝜎 ) |
23 |
22
|
ad2antll |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → 𝜎 ) |
24 |
23 3
|
sylib |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → 𝜂′ ) ) |
25 |
21 24
|
syl5 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) → 𝜂′ ) ) |
26 |
25
|
a2i |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → 𝜂′ ) ) |
27 |
|
pm3.43 |
⊢ ( ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) ∧ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → 𝜂′ ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) |
28 |
26 27
|
mpdan |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) |
29 |
20 28
|
bnj101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) |
30 |
4
|
bnj1247 |
⊢ ( 𝜑0 → 𝑓 ∈ 𝐾 ) |
31 |
30
|
ad2antll |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → 𝑓 ∈ 𝐾 ) |
32 |
|
pm3.43i |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → 𝑓 ∈ 𝐾 ) → ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) ) |
33 |
31 32
|
ax-mp |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) |
34 |
29 33
|
bnj101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) |
35 |
|
fndm |
⊢ ( 𝑓 Fn 𝑛 → dom 𝑓 = 𝑛 ) |
36 |
1 35
|
bnj770 |
⊢ ( 𝜒 → dom 𝑓 = 𝑛 ) |
37 |
36
|
3ad2ant3 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) → dom 𝑓 = 𝑛 ) |
38 |
37
|
ad2antrl |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → dom 𝑓 = 𝑛 ) |
39 |
38
|
eleq2d |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ) |
40 |
|
pm3.43i |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ) → ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) ) ) |
41 |
39 40
|
ax-mp |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) → ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) ) |
42 |
34 41
|
bnj101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) |
43 |
|
bnj268 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ 𝑓 ∈ 𝐾 ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ↔ ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ) |
44 |
|
bnj251 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ 𝑓 ∈ 𝐾 ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ↔ ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) |
45 |
43 44
|
bitr3i |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ↔ ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) |
46 |
45
|
imbi2i |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ) ↔ ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) ) |
47 |
46
|
exbii |
⊢ ( ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ) ↔ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑓 ∈ 𝐾 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ) ) ) ) |
48 |
42 47
|
mpbir |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ) |
49 |
|
simp1 |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) → 𝑗 ∈ 𝑛 ) |
50 |
49
|
bnj706 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → 𝑗 ∈ 𝑛 ) |
51 |
|
simp2 |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) → 𝑖 = suc 𝑗 ) |
52 |
51
|
bnj706 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → 𝑖 = suc 𝑗 ) |
53 |
|
bnj258 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) ↔ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝜂′ ) ∧ 𝑓 ∈ 𝐾 ) ) |
54 |
53
|
simprbi |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → 𝑓 ∈ 𝐾 ) |
55 |
|
bnj642 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ) |
56 |
50 55
|
mpbird |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → 𝑗 ∈ dom 𝑓 ) |
57 |
|
bnj645 |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → 𝜂′ ) |
58 |
57 5
|
sylib |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → ( ( 𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) ) |
59 |
54 56 58
|
mp2and |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) |
60 |
50 52 59
|
3jca |
⊢ ( ( ( 𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖 ) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′ ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) ) |
61 |
48 60
|
bnj1023 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) ) |