Step |
Hyp |
Ref |
Expression |
1 |
|
xpsff1o.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ { 〈 ∅ , 𝑥 〉 , 〈 1o , 𝑦 〉 } ) |
2 |
|
xpsfrnel2 |
⊢ ( { 〈 ∅ , 𝑥 〉 , 〈 1o , 𝑦 〉 } ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
3 |
2
|
biimpri |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → { 〈 ∅ , 𝑥 〉 , 〈 1o , 𝑦 〉 } ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ) |
4 |
3
|
rgen2 |
⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 { 〈 ∅ , 𝑥 〉 , 〈 1o , 𝑦 〉 } ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) |
5 |
1
|
fmpo |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 { 〈 ∅ , 𝑥 〉 , 〈 1o , 𝑦 〉 } ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ 𝐹 : ( 𝐴 × 𝐵 ) ⟶ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ) |
6 |
4 5
|
mpbi |
⊢ 𝐹 : ( 𝐴 × 𝐵 ) ⟶ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) |
7 |
|
1st2nd2 |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → 𝑧 = 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
8 |
7
|
fveq2d |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) ) |
9 |
|
df-ov |
⊢ ( ( 1st ‘ 𝑧 ) 𝐹 ( 2nd ‘ 𝑧 ) ) = ( 𝐹 ‘ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
10 |
|
xp1st |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( 1st ‘ 𝑧 ) ∈ 𝐴 ) |
11 |
|
xp2nd |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( 2nd ‘ 𝑧 ) ∈ 𝐵 ) |
12 |
1
|
xpsfval |
⊢ ( ( ( 1st ‘ 𝑧 ) ∈ 𝐴 ∧ ( 2nd ‘ 𝑧 ) ∈ 𝐵 ) → ( ( 1st ‘ 𝑧 ) 𝐹 ( 2nd ‘ 𝑧 ) ) = { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ) |
13 |
10 11 12
|
syl2anc |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( ( 1st ‘ 𝑧 ) 𝐹 ( 2nd ‘ 𝑧 ) ) = { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ) |
14 |
9 13
|
syl5eqr |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) = { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ) |
15 |
8 14
|
eqtrd |
⊢ ( 𝑧 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 𝑧 ) = { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ) |
16 |
|
1st2nd2 |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → 𝑤 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) |
17 |
16
|
fveq2d |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) ) |
18 |
|
df-ov |
⊢ ( ( 1st ‘ 𝑤 ) 𝐹 ( 2nd ‘ 𝑤 ) ) = ( 𝐹 ‘ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) |
19 |
|
xp1st |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( 1st ‘ 𝑤 ) ∈ 𝐴 ) |
20 |
|
xp2nd |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( 2nd ‘ 𝑤 ) ∈ 𝐵 ) |
21 |
1
|
xpsfval |
⊢ ( ( ( 1st ‘ 𝑤 ) ∈ 𝐴 ∧ ( 2nd ‘ 𝑤 ) ∈ 𝐵 ) → ( ( 1st ‘ 𝑤 ) 𝐹 ( 2nd ‘ 𝑤 ) ) = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ) |
22 |
19 20 21
|
syl2anc |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( ( 1st ‘ 𝑤 ) 𝐹 ( 2nd ‘ 𝑤 ) ) = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ) |
23 |
18 22
|
syl5eqr |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ) |
24 |
17 23
|
eqtrd |
⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → ( 𝐹 ‘ 𝑤 ) = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ) |
25 |
15 24
|
eqeqan12d |
⊢ ( ( 𝑧 ∈ ( 𝐴 × 𝐵 ) ∧ 𝑤 ∈ ( 𝐴 × 𝐵 ) ) → ( ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) ↔ { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ) ) |
26 |
|
fveq1 |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ ∅ ) = ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ ∅ ) ) |
27 |
|
fvex |
⊢ ( 1st ‘ 𝑧 ) ∈ V |
28 |
|
fvpr0o |
⊢ ( ( 1st ‘ 𝑧 ) ∈ V → ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ ∅ ) = ( 1st ‘ 𝑧 ) ) |
29 |
27 28
|
ax-mp |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ ∅ ) = ( 1st ‘ 𝑧 ) |
30 |
|
fvex |
⊢ ( 1st ‘ 𝑤 ) ∈ V |
31 |
|
fvpr0o |
⊢ ( ( 1st ‘ 𝑤 ) ∈ V → ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ ∅ ) = ( 1st ‘ 𝑤 ) ) |
32 |
30 31
|
ax-mp |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ ∅ ) = ( 1st ‘ 𝑤 ) |
33 |
26 29 32
|
3eqtr3g |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → ( 1st ‘ 𝑧 ) = ( 1st ‘ 𝑤 ) ) |
34 |
|
fveq1 |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ 1o ) = ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ 1o ) ) |
35 |
|
fvex |
⊢ ( 2nd ‘ 𝑧 ) ∈ V |
36 |
|
fvpr1o |
⊢ ( ( 2nd ‘ 𝑧 ) ∈ V → ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ 1o ) = ( 2nd ‘ 𝑧 ) ) |
37 |
35 36
|
ax-mp |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } ‘ 1o ) = ( 2nd ‘ 𝑧 ) |
38 |
|
fvex |
⊢ ( 2nd ‘ 𝑤 ) ∈ V |
39 |
|
fvpr1o |
⊢ ( ( 2nd ‘ 𝑤 ) ∈ V → ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ 1o ) = ( 2nd ‘ 𝑤 ) ) |
40 |
38 39
|
ax-mp |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } ‘ 1o ) = ( 2nd ‘ 𝑤 ) |
41 |
34 37 40
|
3eqtr3g |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → ( 2nd ‘ 𝑧 ) = ( 2nd ‘ 𝑤 ) ) |
42 |
33 41
|
opeq12d |
⊢ ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) |
43 |
7 16
|
eqeqan12d |
⊢ ( ( 𝑧 ∈ ( 𝐴 × 𝐵 ) ∧ 𝑤 ∈ ( 𝐴 × 𝐵 ) ) → ( 𝑧 = 𝑤 ↔ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) ) |
44 |
42 43
|
syl5ibr |
⊢ ( ( 𝑧 ∈ ( 𝐴 × 𝐵 ) ∧ 𝑤 ∈ ( 𝐴 × 𝐵 ) ) → ( { 〈 ∅ , ( 1st ‘ 𝑧 ) 〉 , 〈 1o , ( 2nd ‘ 𝑧 ) 〉 } = { 〈 ∅ , ( 1st ‘ 𝑤 ) 〉 , 〈 1o , ( 2nd ‘ 𝑤 ) 〉 } → 𝑧 = 𝑤 ) ) |
45 |
25 44
|
sylbid |
⊢ ( ( 𝑧 ∈ ( 𝐴 × 𝐵 ) ∧ 𝑤 ∈ ( 𝐴 × 𝐵 ) ) → ( ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) → 𝑧 = 𝑤 ) ) |
46 |
45
|
rgen2 |
⊢ ∀ 𝑧 ∈ ( 𝐴 × 𝐵 ) ∀ 𝑤 ∈ ( 𝐴 × 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) → 𝑧 = 𝑤 ) |
47 |
|
dff13 |
⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –1-1→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ∧ ∀ 𝑧 ∈ ( 𝐴 × 𝐵 ) ∀ 𝑤 ∈ ( 𝐴 × 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) → 𝑧 = 𝑤 ) ) ) |
48 |
6 46 47
|
mpbir2an |
⊢ 𝐹 : ( 𝐴 × 𝐵 ) –1-1→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) |
49 |
|
xpsfrnel |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ ( 𝑧 Fn 2o ∧ ( 𝑧 ‘ ∅ ) ∈ 𝐴 ∧ ( 𝑧 ‘ 1o ) ∈ 𝐵 ) ) |
50 |
49
|
simp2bi |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → ( 𝑧 ‘ ∅ ) ∈ 𝐴 ) |
51 |
49
|
simp3bi |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → ( 𝑧 ‘ 1o ) ∈ 𝐵 ) |
52 |
1
|
xpsfval |
⊢ ( ( ( 𝑧 ‘ ∅ ) ∈ 𝐴 ∧ ( 𝑧 ‘ 1o ) ∈ 𝐵 ) → ( ( 𝑧 ‘ ∅ ) 𝐹 ( 𝑧 ‘ 1o ) ) = { 〈 ∅ , ( 𝑧 ‘ ∅ ) 〉 , 〈 1o , ( 𝑧 ‘ 1o ) 〉 } ) |
53 |
50 51 52
|
syl2anc |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → ( ( 𝑧 ‘ ∅ ) 𝐹 ( 𝑧 ‘ 1o ) ) = { 〈 ∅ , ( 𝑧 ‘ ∅ ) 〉 , 〈 1o , ( 𝑧 ‘ 1o ) 〉 } ) |
54 |
|
ixpfn |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → 𝑧 Fn 2o ) |
55 |
|
xpsfeq |
⊢ ( 𝑧 Fn 2o → { 〈 ∅ , ( 𝑧 ‘ ∅ ) 〉 , 〈 1o , ( 𝑧 ‘ 1o ) 〉 } = 𝑧 ) |
56 |
54 55
|
syl |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → { 〈 ∅ , ( 𝑧 ‘ ∅ ) 〉 , 〈 1o , ( 𝑧 ‘ 1o ) 〉 } = 𝑧 ) |
57 |
53 56
|
eqtr2d |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → 𝑧 = ( ( 𝑧 ‘ ∅ ) 𝐹 ( 𝑧 ‘ 1o ) ) ) |
58 |
|
rspceov |
⊢ ( ( ( 𝑧 ‘ ∅ ) ∈ 𝐴 ∧ ( 𝑧 ‘ 1o ) ∈ 𝐵 ∧ 𝑧 = ( ( 𝑧 ‘ ∅ ) 𝐹 ( 𝑧 ‘ 1o ) ) ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 𝐹 𝑏 ) ) |
59 |
50 51 57 58
|
syl3anc |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 𝐹 𝑏 ) ) |
60 |
59
|
rgen |
⊢ ∀ 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 𝐹 𝑏 ) |
61 |
|
foov |
⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ∧ ∀ 𝑧 ∈ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝑧 = ( 𝑎 𝐹 𝑏 ) ) ) |
62 |
6 60 61
|
mpbir2an |
⊢ 𝐹 : ( 𝐴 × 𝐵 ) –onto→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) |
63 |
|
df-f1o |
⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –1-1-onto→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) –1-1→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ∧ 𝐹 : ( 𝐴 × 𝐵 ) –onto→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) ) ) |
64 |
48 62 63
|
mpbir2an |
⊢ 𝐹 : ( 𝐴 × 𝐵 ) –1-1-onto→ X 𝑘 ∈ 2o if ( 𝑘 = ∅ , 𝐴 , 𝐵 ) |