Proof of Theorem infxpenc
Step | Hyp | Ref
| Expression |
1 | | infxpenc.6 |
. . . 4
⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊)) |
2 | | f1ocnv 6149 |
. . . 4
⊢ (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) → ◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → ◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴) |
4 | | infxpenc.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:(ω ↑𝑜
2𝑜)–1-1-onto→ω) |
5 | | f1oi 6174 |
. . . . . . . . 9
⊢ ( I
↾ 𝑊):𝑊–1-1-onto→𝑊 |
6 | 5 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ( I ↾ 𝑊):𝑊–1-1-onto→𝑊) |
7 | | omelon 8543 |
. . . . . . . . . . 11
⊢ ω
∈ On |
8 | 7 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ω ∈
On) |
9 | | 2on 7568 |
. . . . . . . . . 10
⊢
2𝑜 ∈ On |
10 | | oecl 7617 |
. . . . . . . . . 10
⊢ ((ω
∈ On ∧ 2𝑜 ∈ On) → (ω
↑𝑜 2𝑜) ∈ On) |
11 | 8, 9, 10 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → (ω
↑𝑜 2𝑜) ∈ On) |
12 | 9 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 2𝑜
∈ On) |
13 | | peano1 7085 |
. . . . . . . . . . 11
⊢ ∅
∈ ω |
14 | 13 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∅ ∈
ω) |
15 | | oen0 7666 |
. . . . . . . . . 10
⊢
(((ω ∈ On ∧ 2𝑜 ∈ On) ∧
∅ ∈ ω) → ∅ ∈ (ω
↑𝑜 2𝑜)) |
16 | 8, 12, 14, 15 | syl21anc 1325 |
. . . . . . . . 9
⊢ (𝜑 → ∅ ∈ (ω
↑𝑜 2𝑜)) |
17 | | ondif1 7581 |
. . . . . . . . 9
⊢ ((ω
↑𝑜 2𝑜) ∈ (On ∖
1𝑜) ↔ ((ω ↑𝑜
2𝑜) ∈ On ∧ ∅ ∈ (ω
↑𝑜 2𝑜))) |
18 | 11, 16, 17 | sylanbrc 698 |
. . . . . . . 8
⊢ (𝜑 → (ω
↑𝑜 2𝑜) ∈ (On ∖
1𝑜)) |
19 | | infxpenc.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ (On ∖
1𝑜)) |
20 | 19 | eldifad 3586 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ On) |
21 | | infxpenc.5 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘∅) = ∅) |
22 | | infxpenc.k |
. . . . . . . 8
⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑𝑜
2𝑜) ↑𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊)))) |
23 | | infxpenc.h |
. . . . . . . 8
⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑𝑜
2𝑜) CNF 𝑊)) |
24 | 4, 6, 18, 20, 8, 20, 21, 22, 23 | oef1o 8595 |
. . . . . . 7
⊢ (𝜑 → 𝐻:((ω ↑𝑜
2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊)) |
25 | | f1oi 6174 |
. . . . . . . . . 10
⊢ ( I
↾ ω):ω–1-1-onto→ω |
26 | 25 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾
ω):ω–1-1-onto→ω) |
27 | | infxpenc.x |
. . . . . . . . . . 11
⊢ 𝑋 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤)) |
28 | | infxpenc.y |
. . . . . . . . . . 11
⊢ 𝑌 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((2𝑜
·𝑜 𝑤) +𝑜 𝑧)) |
29 | 27, 28 | omf1o 8063 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ On ∧
2𝑜 ∈ On) → (𝑌 ∘ ◡𝑋):(𝑊 ·𝑜
2𝑜)–1-1-onto→(2𝑜
·𝑜 𝑊)) |
30 | 20, 9, 29 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌 ∘ ◡𝑋):(𝑊 ·𝑜
2𝑜)–1-1-onto→(2𝑜
·𝑜 𝑊)) |
31 | | ondif1 7581 |
. . . . . . . . . . 11
⊢ (ω
∈ (On ∖ 1𝑜) ↔ (ω ∈ On ∧
∅ ∈ ω)) |
32 | 7, 13, 31 | mpbir2an 955 |
. . . . . . . . . 10
⊢ ω
∈ (On ∖ 1𝑜) |
33 | 32 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ω ∈ (On ∖
1𝑜)) |
34 | | omcl 7616 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ On ∧
2𝑜 ∈ On) → (𝑊 ·𝑜
2𝑜) ∈ On) |
35 | 20, 9, 34 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → (𝑊 ·𝑜
2𝑜) ∈ On) |
36 | | omcl 7616 |
. . . . . . . . . 10
⊢
((2𝑜 ∈ On ∧ 𝑊 ∈ On) → (2𝑜
·𝑜 𝑊) ∈ On) |
37 | 12, 20, 36 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → (2𝑜
·𝑜 𝑊) ∈ On) |
38 | | fvresi 6439 |
. . . . . . . . . 10
⊢ (∅
∈ ω → (( I ↾ ω)‘∅) =
∅) |
39 | 13, 38 | mp1i 13 |
. . . . . . . . 9
⊢ (𝜑 → (( I ↾
ω)‘∅) = ∅) |
40 | | infxpenc.l |
. . . . . . . . 9
⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚
(𝑊
·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾
ω) ∘ (𝑦 ∘
◡(𝑌 ∘ ◡𝑋)))) |
41 | | infxpenc.j |
. . . . . . . . 9
⊢ 𝐽 = (((ω CNF
(2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·𝑜
2𝑜))) |
42 | 26, 30, 33, 35, 8, 37, 39, 40, 41 | oef1o 8595 |
. . . . . . . 8
⊢ (𝜑 → 𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
(2𝑜 ·𝑜 𝑊))) |
43 | | oeoe 7679 |
. . . . . . . . . 10
⊢ ((ω
∈ On ∧ 2𝑜 ∈ On ∧ 𝑊 ∈ On) → ((ω
↑𝑜 2𝑜) ↑𝑜
𝑊) = (ω
↑𝑜 (2𝑜 ·𝑜
𝑊))) |
44 | 8, 12, 20, 43 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → ((ω
↑𝑜 2𝑜) ↑𝑜
𝑊) = (ω
↑𝑜 (2𝑜 ·𝑜
𝑊))) |
45 | | f1oeq3 6129 |
. . . . . . . . 9
⊢
(((ω ↑𝑜 2𝑜)
↑𝑜 𝑊) = (ω ↑𝑜
(2𝑜 ·𝑜 𝑊)) → (𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→((ω ↑𝑜
2𝑜) ↑𝑜 𝑊) ↔ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
(2𝑜 ·𝑜 𝑊)))) |
46 | 44, 45 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→((ω ↑𝑜
2𝑜) ↑𝑜 𝑊) ↔ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
(2𝑜 ·𝑜 𝑊)))) |
47 | 42, 46 | mpbird 247 |
. . . . . . 7
⊢ (𝜑 → 𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→((ω ↑𝑜
2𝑜) ↑𝑜 𝑊)) |
48 | | f1oco 6159 |
. . . . . . 7
⊢ ((𝐻:((ω
↑𝑜 2𝑜) ↑𝑜
𝑊)–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→((ω ↑𝑜
2𝑜) ↑𝑜 𝑊)) → (𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
𝑊)) |
49 | 24, 47, 48 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
𝑊)) |
50 | | df-2o 7561 |
. . . . . . . . . . . 12
⊢
2𝑜 = suc 1𝑜 |
51 | 50 | oveq2i 6661 |
. . . . . . . . . . 11
⊢ (𝑊 ·𝑜
2𝑜) = (𝑊
·𝑜 suc 1𝑜) |
52 | | 1on 7567 |
. . . . . . . . . . . 12
⊢
1𝑜 ∈ On |
53 | | omsuc 7606 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ On ∧
1𝑜 ∈ On) → (𝑊 ·𝑜 suc
1𝑜) = ((𝑊 ·𝑜
1𝑜) +𝑜 𝑊)) |
54 | 20, 52, 53 | sylancl 694 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 ·𝑜 suc
1𝑜) = ((𝑊 ·𝑜
1𝑜) +𝑜 𝑊)) |
55 | 51, 54 | syl5eq 2668 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑊 ·𝑜
2𝑜) = ((𝑊 ·𝑜
1𝑜) +𝑜 𝑊)) |
56 | | om1 7622 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ On → (𝑊 ·𝑜
1𝑜) = 𝑊) |
57 | 20, 56 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 ·𝑜
1𝑜) = 𝑊) |
58 | 57 | oveq1d 6665 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑊 ·𝑜
1𝑜) +𝑜 𝑊) = (𝑊 +𝑜 𝑊)) |
59 | 55, 58 | eqtrd 2656 |
. . . . . . . . 9
⊢ (𝜑 → (𝑊 ·𝑜
2𝑜) = (𝑊
+𝑜 𝑊)) |
60 | 59 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝜑 → (ω
↑𝑜 (𝑊 ·𝑜
2𝑜)) = (ω ↑𝑜 (𝑊 +𝑜 𝑊))) |
61 | | oeoa 7677 |
. . . . . . . . 9
⊢ ((ω
∈ On ∧ 𝑊 ∈ On
∧ 𝑊 ∈ On) →
(ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜
𝑊)
·𝑜 (ω ↑𝑜 𝑊))) |
62 | 8, 20, 20, 61 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (ω
↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜
𝑊)
·𝑜 (ω ↑𝑜 𝑊))) |
63 | 60, 62 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → (ω
↑𝑜 (𝑊 ·𝑜
2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))) |
64 | | f1oeq2 6128 |
. . . . . . 7
⊢ ((ω
↑𝑜 (𝑊 ·𝑜
2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊)) → ((𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))) |
65 | 63, 64 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜
2𝑜))–1-1-onto→(ω ↑𝑜
𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))) |
66 | 49, 65 | mpbid 222 |
. . . . 5
⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)) |
67 | | oecl 7617 |
. . . . . . 7
⊢ ((ω
∈ On ∧ 𝑊 ∈
On) → (ω ↑𝑜 𝑊) ∈ On) |
68 | 8, 20, 67 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (ω
↑𝑜 𝑊) ∈ On) |
69 | | infxpenc.z |
. . . . . . 7
⊢ 𝑍 = (𝑥 ∈ (ω ↑𝑜
𝑊), 𝑦 ∈ (ω ↑𝑜
𝑊) ↦ (((ω
↑𝑜 𝑊) ·𝑜 𝑥) +𝑜 𝑦)) |
70 | 69 | omxpenlem 8061 |
. . . . . 6
⊢
(((ω ↑𝑜 𝑊) ∈ On ∧ (ω
↑𝑜 𝑊) ∈ On) → 𝑍:((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))) |
71 | 68, 68, 70 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → 𝑍:((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))) |
72 | | f1oco 6159 |
. . . . 5
⊢ (((𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝑍:((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜
(ω ↑𝑜 𝑊))) → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)) |
73 | 66, 71, 72 | syl2anc 693 |
. . . 4
⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)) |
74 | | f1of 6137 |
. . . . . . . . . 10
⊢ (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑𝑜
𝑊)) |
75 | 1, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁:𝐴⟶(ω ↑𝑜
𝑊)) |
76 | 75 | feqmptd 6249 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥))) |
77 | | f1oeq1 6127 |
. . . . . . . 8
⊢ (𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)) → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))) |
78 | 76, 77 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))) |
79 | 1, 78 | mpbid 222 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)) |
80 | 75 | feqmptd 6249 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦))) |
81 | | f1oeq1 6127 |
. . . . . . . 8
⊢ (𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)) → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))) |
82 | 80, 81 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))) |
83 | 1, 82 | mpbid 222 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)) |
84 | 79, 83 | xpf1o 8122 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))) |
85 | | infxpenc.t |
. . . . . 6
⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉) |
86 | | f1oeq1 6127 |
. . . . . 6
⊢ (𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉) → (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊)))) |
87 | 85, 86 | ax-mp 5 |
. . . . 5
⊢ (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))) |
88 | 84, 87 | sylibr 224 |
. . . 4
⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))) |
89 | | f1oco 6159 |
. . . 4
⊢ ((((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω
↑𝑜 𝑊))) → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) |
90 | 73, 88, 89 | syl2anc 693 |
. . 3
⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) |
91 | | f1oco 6159 |
. . 3
⊢ ((◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴 ∧ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴) |
92 | 3, 90, 91 | syl2anc 693 |
. 2
⊢ (𝜑 → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴) |
93 | | infxpenc.g |
. . 3
⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) |
94 | | f1oeq1 6127 |
. . 3
⊢ (𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)) |
95 | 93, 94 | ax-mp 5 |
. 2
⊢ (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴) |
96 | 92, 95 | sylibr 224 |
1
⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴) |