Step | Hyp | Ref
| Expression |
1 | | xpsfrnel2 17447 |
. . . . . 6
⊢
({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
2 | 1 | biimpri 227 |
. . . . 5
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵)) |
3 | 2 | rgen2 3195 |
. . . 4
⊢
∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) |
4 | | xpsff1o.f |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) |
5 | 4 | fmpo 8001 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)) |
6 | 3, 5 | mpbi 229 |
. . 3
⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
7 | | 1st2nd2 7961 |
. . . . . . . 8
⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) |
8 | 7 | fveq2d 6847 |
. . . . . . 7
⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)) |
9 | | df-ov 7361 |
. . . . . . . 8
⊢
((1st ‘𝑧)𝐹(2nd ‘𝑧)) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) |
10 | | xp1st 7954 |
. . . . . . . . 9
⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1st ‘𝑧) ∈ 𝐴) |
11 | | xp2nd 7955 |
. . . . . . . . 9
⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2nd ‘𝑧) ∈ 𝐵) |
12 | 4 | xpsfval 17449 |
. . . . . . . . 9
⊢
(((1st ‘𝑧) ∈ 𝐴 ∧ (2nd ‘𝑧) ∈ 𝐵) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = {⟨∅, (1st
‘𝑧)⟩,
⟨1o, (2nd ‘𝑧)⟩}) |
13 | 10, 11, 12 | syl2anc 585 |
. . . . . . . 8
⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = {⟨∅, (1st
‘𝑧)⟩,
⟨1o, (2nd ‘𝑧)⟩}) |
14 | 9, 13 | eqtr3id 2791 |
. . . . . . 7
⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) = {⟨∅,
(1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}) |
15 | 8, 14 | eqtrd 2777 |
. . . . . 6
⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = {⟨∅, (1st
‘𝑧)⟩,
⟨1o, (2nd ‘𝑧)⟩}) |
16 | | 1st2nd2 7961 |
. . . . . . . 8
⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩) |
17 | 16 | fveq2d 6847 |
. . . . . . 7
⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)) |
18 | | df-ov 7361 |
. . . . . . . 8
⊢
((1st ‘𝑤)𝐹(2nd ‘𝑤)) = (𝐹‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩) |
19 | | xp1st 7954 |
. . . . . . . . 9
⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1st ‘𝑤) ∈ 𝐴) |
20 | | xp2nd 7955 |
. . . . . . . . 9
⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2nd ‘𝑤) ∈ 𝐵) |
21 | 4 | xpsfval 17449 |
. . . . . . . . 9
⊢
(((1st ‘𝑤) ∈ 𝐴 ∧ (2nd ‘𝑤) ∈ 𝐵) → ((1st ‘𝑤)𝐹(2nd ‘𝑤)) = {⟨∅, (1st
‘𝑤)⟩,
⟨1o, (2nd ‘𝑤)⟩}) |
22 | 19, 20, 21 | syl2anc 585 |
. . . . . . . 8
⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1st ‘𝑤)𝐹(2nd ‘𝑤)) = {⟨∅, (1st
‘𝑤)⟩,
⟨1o, (2nd ‘𝑤)⟩}) |
23 | 18, 22 | eqtr3id 2791 |
. . . . . . 7
⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩) = {⟨∅,
(1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}) |
24 | 17, 23 | eqtrd 2777 |
. . . . . 6
⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = {⟨∅, (1st
‘𝑤)⟩,
⟨1o, (2nd ‘𝑤)⟩}) |
25 | 15, 24 | eqeqan12d 2751 |
. . . . 5
⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {⟨∅, (1st
‘𝑧)⟩,
⟨1o, (2nd ‘𝑧)⟩} = {⟨∅, (1st
‘𝑤)⟩,
⟨1o, (2nd ‘𝑤)⟩})) |
26 | | fveq1 6842 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩} =
{⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩} →
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘∅) = ({⟨∅,
(1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘∅)) |
27 | | fvex 6856 |
. . . . . . . . 9
⊢
(1st ‘𝑧) ∈ V |
28 | | fvpr0o 17442 |
. . . . . . . . 9
⊢
((1st ‘𝑧) ∈ V → ({⟨∅,
(1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘∅) = (1st
‘𝑧)) |
29 | 27, 28 | ax-mp 5 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘∅) = (1st
‘𝑧) |
30 | | fvex 6856 |
. . . . . . . . 9
⊢
(1st ‘𝑤) ∈ V |
31 | | fvpr0o 17442 |
. . . . . . . . 9
⊢
((1st ‘𝑤) ∈ V → ({⟨∅,
(1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘∅) = (1st
‘𝑤)) |
32 | 30, 31 | ax-mp 5 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘∅) = (1st
‘𝑤) |
33 | 26, 29, 32 | 3eqtr3g 2800 |
. . . . . . 7
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩} =
{⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩} →
(1st ‘𝑧) =
(1st ‘𝑤)) |
34 | | fveq1 6842 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩} =
{⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩} →
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘1o) =
({⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘1o)) |
35 | | fvex 6856 |
. . . . . . . . 9
⊢
(2nd ‘𝑧) ∈ V |
36 | | fvpr1o 17443 |
. . . . . . . . 9
⊢
((2nd ‘𝑧) ∈ V → ({⟨∅,
(1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘1o) =
(2nd ‘𝑧)) |
37 | 35, 36 | ax-mp 5 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩}‘1o) =
(2nd ‘𝑧) |
38 | | fvex 6856 |
. . . . . . . . 9
⊢
(2nd ‘𝑤) ∈ V |
39 | | fvpr1o 17443 |
. . . . . . . . 9
⊢
((2nd ‘𝑤) ∈ V → ({⟨∅,
(1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘1o) =
(2nd ‘𝑤)) |
40 | 38, 39 | ax-mp 5 |
. . . . . . . 8
⊢
({⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩}‘1o) =
(2nd ‘𝑤) |
41 | 34, 37, 40 | 3eqtr3g 2800 |
. . . . . . 7
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩} =
{⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩} →
(2nd ‘𝑧) =
(2nd ‘𝑤)) |
42 | 33, 41 | opeq12d 4839 |
. . . . . 6
⊢
({⟨∅, (1st ‘𝑧)⟩, ⟨1o, (2nd
‘𝑧)⟩} =
{⟨∅, (1st ‘𝑤)⟩, ⟨1o, (2nd
‘𝑤)⟩} →
⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩) |
43 | 7, 16 | eqeqan12d 2751 |
. . . . . 6
⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st
‘𝑤), (2nd
‘𝑤)⟩)) |
44 | 42, 43 | syl5ibr 246 |
. . . . 5
⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1st
‘𝑧)⟩,
⟨1o, (2nd ‘𝑧)⟩} = {⟨∅, (1st
‘𝑤)⟩,
⟨1o, (2nd ‘𝑤)⟩} → 𝑧 = 𝑤)) |
45 | 25, 44 | sylbid 239 |
. . . 4
⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
46 | 45 | rgen2 3195 |
. . 3
⊢
∀𝑧 ∈
(𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
47 | | dff13 7203 |
. . 3
⊢ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))) |
48 | 6, 46, 47 | mpbir2an 710 |
. 2
⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
49 | | xpsfrnel 17445 |
. . . . . 6
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵)) |
50 | 49 | simp2bi 1147 |
. . . . 5
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴) |
51 | 49 | simp3bi 1148 |
. . . . 5
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → (𝑧‘1o) ∈ 𝐵) |
52 | 4 | xpsfval 17449 |
. . . . . . 7
⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅,
(𝑧‘∅)⟩,
⟨1o, (𝑧‘1o)⟩}) |
53 | 50, 51, 52 | syl2anc 585 |
. . . . . 6
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅,
(𝑧‘∅)⟩,
⟨1o, (𝑧‘1o)⟩}) |
54 | | ixpfn 8842 |
. . . . . . 7
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → 𝑧 Fn 2o) |
55 | | xpsfeq 17446 |
. . . . . . 7
⊢ (𝑧 Fn 2o →
{⟨∅, (𝑧‘∅)⟩, ⟨1o,
(𝑧‘1o)⟩} = 𝑧) |
56 | 54, 55 | syl 17 |
. . . . . 6
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → {⟨∅, (𝑧‘∅)⟩,
⟨1o, (𝑧‘1o)⟩} = 𝑧) |
57 | 53, 56 | eqtr2d 2778 |
. . . . 5
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) |
58 | | rspceov 7405 |
. . . . 5
⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵 ∧ 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)) |
59 | 50, 51, 57, 58 | syl3anc 1372 |
. . . 4
⊢ (𝑧 ∈ X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)) |
60 | 59 | rgen 3067 |
. . 3
⊢
∀𝑧 ∈
X 𝑘
∈ 2o if(𝑘 =
∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏) |
61 | | foov 7529 |
. . 3
⊢ (𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ X 𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))) |
62 | 6, 60, 61 | mpbir2an 710 |
. 2
⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
63 | | df-f1o 6504 |
. 2
⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ∧ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))) |
64 | 48, 62, 63 | mpbir2an 710 |
1
⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈
2o if(𝑘 =
∅, 𝐴, 𝐵) |