Step |
Hyp |
Ref |
Expression |
1 |
|
cantnfs.s |
⊢ 𝑆 = dom ( 𝐴 CNF 𝐵 ) |
2 |
|
cantnfs.a |
⊢ ( 𝜑 → 𝐴 ∈ On ) |
3 |
|
cantnfs.b |
⊢ ( 𝜑 → 𝐵 ∈ On ) |
4 |
|
oemapval.t |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑧 ∈ 𝐵 ( ( 𝑥 ‘ 𝑧 ) ∈ ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐵 ( 𝑧 ∈ 𝑤 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) } |
5 |
|
cantnf.c |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ↑o 𝐵 ) ) |
6 |
|
cantnf.s |
⊢ ( 𝜑 → 𝐶 ⊆ ran ( 𝐴 CNF 𝐵 ) ) |
7 |
|
cantnf.e |
⊢ ( 𝜑 → ∅ ∈ 𝐶 ) |
8 |
|
cantnf.x |
⊢ 𝑋 = ∪ ∩ { 𝑐 ∈ On ∣ 𝐶 ∈ ( 𝐴 ↑o 𝑐 ) } |
9 |
|
cantnf.p |
⊢ 𝑃 = ( ℩ 𝑑 ∃ 𝑎 ∈ On ∃ 𝑏 ∈ ( 𝐴 ↑o 𝑋 ) ( 𝑑 = 〈 𝑎 , 𝑏 〉 ∧ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑎 ) +o 𝑏 ) = 𝐶 ) ) |
10 |
|
cantnf.y |
⊢ 𝑌 = ( 1st ‘ 𝑃 ) |
11 |
|
cantnf.z |
⊢ 𝑍 = ( 2nd ‘ 𝑃 ) |
12 |
1 2 3 4 5 6 7
|
cantnflem2 |
⊢ ( 𝜑 → ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝐶 ∈ ( On ∖ 1o ) ) ) |
13 |
|
eqid |
⊢ 𝑋 = 𝑋 |
14 |
|
eqid |
⊢ 𝑌 = 𝑌 |
15 |
|
eqid |
⊢ 𝑍 = 𝑍 |
16 |
13 14 15
|
3pm3.2i |
⊢ ( 𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍 ) |
17 |
8 9 10 11
|
oeeui |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝐶 ∈ ( On ∖ 1o ) ) → ( ( ( 𝑋 ∈ On ∧ 𝑌 ∈ ( 𝐴 ∖ 1o ) ∧ 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) ∧ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) = 𝐶 ) ↔ ( 𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍 ) ) ) |
18 |
16 17
|
mpbiri |
⊢ ( ( 𝐴 ∈ ( On ∖ 2o ) ∧ 𝐶 ∈ ( On ∖ 1o ) ) → ( ( 𝑋 ∈ On ∧ 𝑌 ∈ ( 𝐴 ∖ 1o ) ∧ 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) ∧ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) = 𝐶 ) ) |
19 |
12 18
|
syl |
⊢ ( 𝜑 → ( ( 𝑋 ∈ On ∧ 𝑌 ∈ ( 𝐴 ∖ 1o ) ∧ 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) ∧ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) = 𝐶 ) ) |
20 |
19
|
simpld |
⊢ ( 𝜑 → ( 𝑋 ∈ On ∧ 𝑌 ∈ ( 𝐴 ∖ 1o ) ∧ 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) ) |
21 |
20
|
simp1d |
⊢ ( 𝜑 → 𝑋 ∈ On ) |
22 |
|
oecl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑋 ∈ On ) → ( 𝐴 ↑o 𝑋 ) ∈ On ) |
23 |
2 21 22
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ↑o 𝑋 ) ∈ On ) |
24 |
20
|
simp2d |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐴 ∖ 1o ) ) |
25 |
24
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝐴 ) |
26 |
|
onelon |
⊢ ( ( 𝐴 ∈ On ∧ 𝑌 ∈ 𝐴 ) → 𝑌 ∈ On ) |
27 |
2 25 26
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ∈ On ) |
28 |
|
omcl |
⊢ ( ( ( 𝐴 ↑o 𝑋 ) ∈ On ∧ 𝑌 ∈ On ) → ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ∈ On ) |
29 |
23 27 28
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ∈ On ) |
30 |
20
|
simp3d |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) |
31 |
|
onelon |
⊢ ( ( ( 𝐴 ↑o 𝑋 ) ∈ On ∧ 𝑍 ∈ ( 𝐴 ↑o 𝑋 ) ) → 𝑍 ∈ On ) |
32 |
23 30 31
|
syl2anc |
⊢ ( 𝜑 → 𝑍 ∈ On ) |
33 |
|
oaword1 |
⊢ ( ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ∈ On ∧ 𝑍 ∈ On ) → ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ⊆ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) ) |
34 |
29 32 33
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ⊆ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) ) |
35 |
|
dif1o |
⊢ ( 𝑌 ∈ ( 𝐴 ∖ 1o ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅ ) ) |
36 |
35
|
simprbi |
⊢ ( 𝑌 ∈ ( 𝐴 ∖ 1o ) → 𝑌 ≠ ∅ ) |
37 |
24 36
|
syl |
⊢ ( 𝜑 → 𝑌 ≠ ∅ ) |
38 |
|
on0eln0 |
⊢ ( 𝑌 ∈ On → ( ∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅ ) ) |
39 |
27 38
|
syl |
⊢ ( 𝜑 → ( ∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅ ) ) |
40 |
37 39
|
mpbird |
⊢ ( 𝜑 → ∅ ∈ 𝑌 ) |
41 |
|
omword1 |
⊢ ( ( ( ( 𝐴 ↑o 𝑋 ) ∈ On ∧ 𝑌 ∈ On ) ∧ ∅ ∈ 𝑌 ) → ( 𝐴 ↑o 𝑋 ) ⊆ ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ) |
42 |
23 27 40 41
|
syl21anc |
⊢ ( 𝜑 → ( 𝐴 ↑o 𝑋 ) ⊆ ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ) |
43 |
42 30
|
sseldd |
⊢ ( 𝜑 → 𝑍 ∈ ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) ) |
44 |
34 43
|
sseldd |
⊢ ( 𝜑 → 𝑍 ∈ ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) ) |
45 |
19
|
simprd |
⊢ ( 𝜑 → ( ( ( 𝐴 ↑o 𝑋 ) ·o 𝑌 ) +o 𝑍 ) = 𝐶 ) |
46 |
44 45
|
eleqtrd |
⊢ ( 𝜑 → 𝑍 ∈ 𝐶 ) |
47 |
6 46
|
sseldd |
⊢ ( 𝜑 → 𝑍 ∈ ran ( 𝐴 CNF 𝐵 ) ) |
48 |
1 2 3
|
cantnff |
⊢ ( 𝜑 → ( 𝐴 CNF 𝐵 ) : 𝑆 ⟶ ( 𝐴 ↑o 𝐵 ) ) |
49 |
|
ffn |
⊢ ( ( 𝐴 CNF 𝐵 ) : 𝑆 ⟶ ( 𝐴 ↑o 𝐵 ) → ( 𝐴 CNF 𝐵 ) Fn 𝑆 ) |
50 |
|
fvelrnb |
⊢ ( ( 𝐴 CNF 𝐵 ) Fn 𝑆 → ( 𝑍 ∈ ran ( 𝐴 CNF 𝐵 ) ↔ ∃ 𝑔 ∈ 𝑆 ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) |
51 |
48 49 50
|
3syl |
⊢ ( 𝜑 → ( 𝑍 ∈ ran ( 𝐴 CNF 𝐵 ) ↔ ∃ 𝑔 ∈ 𝑆 ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) |
52 |
47 51
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝑆 ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) |
53 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝐴 ∈ On ) |
54 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝐵 ∈ On ) |
55 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝐶 ∈ ( 𝐴 ↑o 𝐵 ) ) |
56 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝐶 ⊆ ran ( 𝐴 CNF 𝐵 ) ) |
57 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → ∅ ∈ 𝐶 ) |
58 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝑔 ∈ 𝑆 ) |
59 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) |
60 |
|
eqid |
⊢ ( 𝑡 ∈ 𝐵 ↦ if ( 𝑡 = 𝑋 , 𝑌 , ( 𝑔 ‘ 𝑡 ) ) ) = ( 𝑡 ∈ 𝐵 ↦ if ( 𝑡 = 𝑋 , 𝑌 , ( 𝑔 ‘ 𝑡 ) ) ) |
61 |
1 53 54 4 55 56 57 8 9 10 11 58 59 60
|
cantnflem3 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝑆 ∧ ( ( 𝐴 CNF 𝐵 ) ‘ 𝑔 ) = 𝑍 ) ) → 𝐶 ∈ ran ( 𝐴 CNF 𝐵 ) ) |
62 |
52 61
|
rexlimddv |
⊢ ( 𝜑 → 𝐶 ∈ ran ( 𝐴 CNF 𝐵 ) ) |