Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1128.1 |
⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
2 |
|
bnj1128.2 |
⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
3 |
|
bnj1128.3 |
⊢ 𝐷 = ( ω ∖ { ∅ } ) |
4 |
|
bnj1128.4 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } |
5 |
|
bnj1128.5 |
⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
6 |
|
bnj1128.6 |
⊢ ( 𝜃 ↔ ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) |
7 |
|
bnj1128.7 |
⊢ ( 𝜏 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜃 ) ) |
8 |
|
bnj1128.8 |
⊢ ( 𝜑′ ↔ [ 𝑗 / 𝑖 ] 𝜑 ) |
9 |
|
bnj1128.9 |
⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 ) |
10 |
|
bnj1128.10 |
⊢ ( 𝜒′ ↔ [ 𝑗 / 𝑖 ] 𝜒 ) |
11 |
|
bnj1128.11 |
⊢ ( 𝜃′ ↔ [ 𝑗 / 𝑖 ] 𝜃 ) |
12 |
1 2 3 4 5
|
bnj981 |
⊢ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) ) |
13 |
|
simp1 |
⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → 𝜒 ) |
14 |
|
simp2 |
⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → 𝑖 ∈ 𝑛 ) |
15 |
|
nfv |
⊢ Ⅎ 𝑗 𝑖 ∈ 𝑛 |
16 |
|
nfra1 |
⊢ Ⅎ 𝑗 ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜃 ) |
17 |
7 16
|
nfxfr |
⊢ Ⅎ 𝑗 𝜏 |
18 |
|
nfv |
⊢ Ⅎ 𝑗 𝜒 |
19 |
15 17 18
|
nf3an |
⊢ Ⅎ 𝑗 ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) |
20 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 |
21 |
19 20
|
nfim |
⊢ Ⅎ 𝑗 ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
22 |
21
|
nf5ri |
⊢ ( ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) → ∀ 𝑗 ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) |
23 |
3
|
bnj1098 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) |
24 |
|
simpl |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → 𝑖 ≠ ∅ ) |
25 |
|
simpr1 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → 𝑖 ∈ 𝑛 ) |
26 |
5
|
bnj1232 |
⊢ ( 𝜒 → 𝑛 ∈ 𝐷 ) |
27 |
26
|
3ad2ant3 |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → 𝑛 ∈ 𝐷 ) |
28 |
27
|
adantl |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → 𝑛 ∈ 𝐷 ) |
29 |
24 25 28
|
3jca |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) ) |
30 |
23 29
|
bnj1101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) |
31 |
|
ancl |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) ) |
32 |
30 31
|
bnj101 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) |
33 |
|
df-3an |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ↔ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) |
34 |
33
|
imbi2i |
⊢ ( ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) ↔ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) ) |
35 |
34
|
exbii |
⊢ ( ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) ↔ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) ) |
36 |
32 35
|
mpbir |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) ) |
37 |
|
bnj213 |
⊢ pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ 𝐴 |
38 |
37
|
bnj226 |
⊢ ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ 𝐴 |
39 |
|
simp21 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝑖 ∈ 𝑛 ) |
40 |
|
simp3r |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝑖 = suc 𝑗 ) |
41 |
|
biid |
⊢ ( 𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷 ) |
42 |
|
biid |
⊢ ( 𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛 ) |
43 |
|
vex |
⊢ 𝑗 ∈ V |
44 |
|
sbcg |
⊢ ( 𝑗 ∈ V → ( [ 𝑗 / 𝑖 ] 𝜑 ↔ 𝜑 ) ) |
45 |
43 44
|
ax-mp |
⊢ ( [ 𝑗 / 𝑖 ] 𝜑 ↔ 𝜑 ) |
46 |
8 45
|
bitr2i |
⊢ ( 𝜑 ↔ 𝜑′ ) |
47 |
2 9
|
bnj1039 |
⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
48 |
2 47
|
bitr4i |
⊢ ( 𝜓 ↔ 𝜓′ ) |
49 |
41 42 46 48
|
bnj887 |
⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) ) |
50 |
8 9 5 10
|
bnj1040 |
⊢ ( 𝜒′ ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) ) |
51 |
49 5 50
|
3bitr4i |
⊢ ( 𝜒 ↔ 𝜒′ ) |
52 |
50
|
bnj1254 |
⊢ ( 𝜒′ → 𝜓′ ) |
53 |
51 52
|
sylbi |
⊢ ( 𝜒 → 𝜓′ ) |
54 |
53
|
3ad2ant3 |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → 𝜓′ ) |
55 |
54
|
3ad2ant2 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝜓′ ) |
56 |
|
simp3l |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝑗 ∈ 𝑛 ) |
57 |
27
|
3ad2ant2 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝑛 ∈ 𝐷 ) |
58 |
3
|
bnj923 |
⊢ ( 𝑛 ∈ 𝐷 → 𝑛 ∈ ω ) |
59 |
|
elnn |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω ) → 𝑗 ∈ ω ) |
60 |
58 59
|
sylan2 |
⊢ ( ( 𝑗 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ) → 𝑗 ∈ ω ) |
61 |
56 57 60
|
syl2anc |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → 𝑗 ∈ ω ) |
62 |
47
|
bnj589 |
⊢ ( 𝜓′ ↔ ∀ 𝑗 ∈ ω ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
63 |
|
rsp |
⊢ ( ∀ 𝑗 ∈ ω ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) → ( 𝑗 ∈ ω → ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
64 |
62 63
|
sylbi |
⊢ ( 𝜓′ → ( 𝑗 ∈ ω → ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
65 |
|
eleq1 |
⊢ ( 𝑖 = suc 𝑗 → ( 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛 ) ) |
66 |
|
fveqeq2 |
⊢ ( 𝑖 = suc 𝑗 → ( ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
67 |
65 66
|
imbi12d |
⊢ ( 𝑖 = suc 𝑗 → ( ( 𝑖 ∈ 𝑛 → ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
68 |
67
|
imbi2d |
⊢ ( 𝑖 = suc 𝑗 → ( ( 𝑗 ∈ ω → ( 𝑖 ∈ 𝑛 → ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑗 ∈ ω → ( suc 𝑗 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) ) |
69 |
64 68
|
syl5ibr |
⊢ ( 𝑖 = suc 𝑗 → ( 𝜓′ → ( 𝑗 ∈ ω → ( 𝑖 ∈ 𝑛 → ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) ) |
70 |
40 55 61 69
|
syl3c |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → ( 𝑖 ∈ 𝑛 → ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
71 |
39 70
|
mpd |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → ( 𝑓 ‘ 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) |
72 |
38 71
|
bnj1262 |
⊢ ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ∧ ( 𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
73 |
36 72
|
bnj1023 |
⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
74 |
5
|
bnj1247 |
⊢ ( 𝜒 → 𝜑 ) |
75 |
74
|
3ad2ant3 |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → 𝜑 ) |
76 |
|
bnj213 |
⊢ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐴 |
77 |
|
fveq2 |
⊢ ( 𝑖 = ∅ → ( 𝑓 ‘ 𝑖 ) = ( 𝑓 ‘ ∅ ) ) |
78 |
1
|
biimpi |
⊢ ( 𝜑 → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
79 |
77 78
|
sylan9eq |
⊢ ( ( 𝑖 = ∅ ∧ 𝜑 ) → ( 𝑓 ‘ 𝑖 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
80 |
76 79
|
bnj1262 |
⊢ ( ( 𝑖 = ∅ ∧ 𝜑 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
81 |
75 80
|
sylan2 |
⊢ ( ( 𝑖 = ∅ ∧ ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
82 |
73 81
|
bnj1109 |
⊢ ∃ 𝑗 ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
83 |
22 82
|
bnj1131 |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
84 |
83
|
3expia |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ) → ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) |
85 |
84 6
|
sylibr |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝜏 ) → 𝜃 ) |
86 |
3 5 7 85
|
bnj1133 |
⊢ ( 𝜒 → ∀ 𝑖 ∈ 𝑛 𝜃 ) |
87 |
6
|
ralbii |
⊢ ( ∀ 𝑖 ∈ 𝑛 𝜃 ↔ ∀ 𝑖 ∈ 𝑛 ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) |
88 |
86 87
|
sylib |
⊢ ( 𝜒 → ∀ 𝑖 ∈ 𝑛 ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) |
89 |
|
rsp |
⊢ ( ∀ 𝑖 ∈ 𝑛 ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) → ( 𝑖 ∈ 𝑛 → ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) ) |
90 |
88 89
|
syl |
⊢ ( 𝜒 → ( 𝑖 ∈ 𝑛 → ( 𝜒 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) ) ) |
91 |
13 14 13 90
|
syl3c |
⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐴 ) |
92 |
|
simp3 |
⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) |
93 |
91 92
|
sseldd |
⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → 𝑌 ∈ 𝐴 ) |
94 |
93
|
2eximi |
⊢ ( ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ ( 𝑓 ‘ 𝑖 ) ) → ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ) |
95 |
12 94
|
bnj593 |
⊢ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ) |
96 |
|
19.9v |
⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ↔ ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ) |
97 |
|
19.9v |
⊢ ( ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ↔ ∃ 𝑖 𝑌 ∈ 𝐴 ) |
98 |
|
19.9v |
⊢ ( ∃ 𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴 ) |
99 |
96 97 98
|
3bitri |
⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴 ) |
100 |
95 99
|
sylib |
⊢ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → 𝑌 ∈ 𝐴 ) |