Step |
Hyp |
Ref |
Expression |
1 |
|
projf1o.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
projf1o.2 |
⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ 〈 𝐴 , 𝑥 〉 ) |
3 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ { 𝐴 } ) |
5 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐴 ∈ { 𝐴 } ) |
6 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
7 |
5 6
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 〈 𝐴 , 𝑦 〉 ∈ ( { 𝐴 } × 𝐵 ) ) |
8 |
|
opeq2 |
⊢ ( 𝑥 = 𝑦 → 〈 𝐴 , 𝑥 〉 = 〈 𝐴 , 𝑦 〉 ) |
9 |
8
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝐵 ↦ 〈 𝐴 , 𝑥 〉 ) = ( 𝑦 ∈ 𝐵 ↦ 〈 𝐴 , 𝑦 〉 ) |
10 |
2 9
|
eqtri |
⊢ 𝐹 = ( 𝑦 ∈ 𝐵 ↦ 〈 𝐴 , 𝑦 〉 ) |
11 |
7 10
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ ( { 𝐴 } × 𝐵 ) ) |
12 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → 𝜑 ) |
13 |
2 8 6 7
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑦 ) = 〈 𝐴 , 𝑦 〉 ) |
14 |
13
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 〈 𝐴 , 𝑦 〉 = ( 𝐹 ‘ 𝑦 ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 〈 𝐴 , 𝑦 〉 = ( 𝐹 ‘ 𝑦 ) ) |
16 |
15
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → 〈 𝐴 , 𝑦 〉 = ( 𝐹 ‘ 𝑦 ) ) |
17 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
18 |
|
opeq2 |
⊢ ( 𝑦 = 𝑧 → 〈 𝐴 , 𝑦 〉 = 〈 𝐴 , 𝑧 〉 ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝐵 ) |
20 |
|
opex |
⊢ 〈 𝐴 , 𝑧 〉 ∈ V |
21 |
20
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 〈 𝐴 , 𝑧 〉 ∈ V ) |
22 |
10 18 19 21
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑧 ) = 〈 𝐴 , 𝑧 〉 ) |
23 |
22
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑧 ) = 〈 𝐴 , 𝑧 〉 ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → ( 𝐹 ‘ 𝑧 ) = 〈 𝐴 , 𝑧 〉 ) |
25 |
16 17 24
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → 〈 𝐴 , 𝑦 〉 = 〈 𝐴 , 𝑧 〉 ) |
26 |
|
vex |
⊢ 𝑧 ∈ V |
27 |
26
|
a1i |
⊢ ( 𝜑 → 𝑧 ∈ V ) |
28 |
|
opthg2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V ) → ( 〈 𝐴 , 𝑦 〉 = 〈 𝐴 , 𝑧 〉 ↔ ( 𝐴 = 𝐴 ∧ 𝑦 = 𝑧 ) ) ) |
29 |
1 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 〈 𝐴 , 𝑦 〉 = 〈 𝐴 , 𝑧 〉 ↔ ( 𝐴 = 𝐴 ∧ 𝑦 = 𝑧 ) ) ) |
30 |
29
|
simplbda |
⊢ ( ( 𝜑 ∧ 〈 𝐴 , 𝑦 〉 = 〈 𝐴 , 𝑧 〉 ) → 𝑦 = 𝑧 ) |
31 |
12 25 30
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) → 𝑦 = 𝑧 ) |
32 |
31
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
33 |
32
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
34 |
33
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
35 |
|
dff13 |
⊢ ( 𝐹 : 𝐵 –1-1→ ( { 𝐴 } × 𝐵 ) ↔ ( 𝐹 : 𝐵 ⟶ ( { 𝐴 } × 𝐵 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
36 |
11 34 35
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐵 –1-1→ ( { 𝐴 } × 𝐵 ) ) |
37 |
|
elsnxp |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 𝑧 = 〈 𝐴 , 𝑦 〉 ) ) |
38 |
1 37
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 𝑧 = 〈 𝐴 , 𝑦 〉 ) ) |
39 |
38
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ) → ∃ 𝑦 ∈ 𝐵 𝑧 = 〈 𝐴 , 𝑦 〉 ) |
40 |
13
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝐴 , 𝑦 〉 ) → ( 𝐹 ‘ 𝑦 ) = 〈 𝐴 , 𝑦 〉 ) |
41 |
|
id |
⊢ ( 𝑧 = 〈 𝐴 , 𝑦 〉 → 𝑧 = 〈 𝐴 , 𝑦 〉 ) |
42 |
41
|
eqcomd |
⊢ ( 𝑧 = 〈 𝐴 , 𝑦 〉 → 〈 𝐴 , 𝑦 〉 = 𝑧 ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝐴 , 𝑦 〉 ) → 〈 𝐴 , 𝑦 〉 = 𝑧 ) |
44 |
40 43
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = 〈 𝐴 , 𝑦 〉 ) → 𝑧 = ( 𝐹 ‘ 𝑦 ) ) |
45 |
44
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 = 〈 𝐴 , 𝑦 〉 → 𝑧 = ( 𝐹 ‘ 𝑦 ) ) ) |
46 |
45
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 = 〈 𝐴 , 𝑦 〉 → 𝑧 = ( 𝐹 ‘ 𝑦 ) ) ) |
47 |
46
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ) → ( ∃ 𝑦 ∈ 𝐵 𝑧 = 〈 𝐴 , 𝑦 〉 → ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝐹 ‘ 𝑦 ) ) ) |
48 |
39 47
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ) → ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝐹 ‘ 𝑦 ) ) |
49 |
48
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝐹 ‘ 𝑦 ) ) |
50 |
|
dffo3 |
⊢ ( 𝐹 : 𝐵 –onto→ ( { 𝐴 } × 𝐵 ) ↔ ( 𝐹 : 𝐵 ⟶ ( { 𝐴 } × 𝐵 ) ∧ ∀ 𝑧 ∈ ( { 𝐴 } × 𝐵 ) ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝐹 ‘ 𝑦 ) ) ) |
51 |
11 49 50
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐵 –onto→ ( { 𝐴 } × 𝐵 ) ) |
52 |
|
df-f1o |
⊢ ( 𝐹 : 𝐵 –1-1-onto→ ( { 𝐴 } × 𝐵 ) ↔ ( 𝐹 : 𝐵 –1-1→ ( { 𝐴 } × 𝐵 ) ∧ 𝐹 : 𝐵 –onto→ ( { 𝐴 } × 𝐵 ) ) ) |
53 |
36 51 52
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐵 –1-1-onto→ ( { 𝐴 } × 𝐵 ) ) |