Step | Hyp | Ref
| Expression |
1 | | sneq 3409 |
. . . 4
⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) |
2 | | f1oeq2 5138 |
. . . 4
⊢ ({𝑎} = {𝐴} → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
3 | 1, 2 | syl 14 |
. . 3
⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
4 | | opeq1 3570 |
. . . . 5
⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) |
5 | 4 | sneqd 3411 |
. . . 4
⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
6 | | f1oeq1 5137 |
. . . 4
⊢
({〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉} → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
7 | 5, 6 | syl 14 |
. . 3
⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
8 | 3, 7 | bitrd 186 |
. 2
⊢ (𝑎 = 𝐴 → ({〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏})) |
9 | | sneq 3409 |
. . . 4
⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) |
10 | | f1oeq3 5139 |
. . . 4
⊢ ({𝑏} = {𝐵} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
11 | 9, 10 | syl 14 |
. . 3
⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵})) |
12 | | opeq2 3571 |
. . . . 5
⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) |
13 | 12 | sneqd 3411 |
. . . 4
⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
14 | | f1oeq1 5137 |
. . . 4
⊢
({〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉} → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
15 | 13, 14 | syl 14 |
. . 3
⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
16 | 11, 15 | bitrd 186 |
. 2
⊢ (𝑏 = 𝐵 → ({〈𝐴, 𝑏〉}:{𝐴}–1-1-onto→{𝑏} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) |
17 | | vex 2604 |
. . 3
⊢ 𝑎 ∈ V |
18 | | vex 2604 |
. . 3
⊢ 𝑏 ∈ V |
19 | 17, 18 | f1osn 5186 |
. 2
⊢
{〈𝑎, 𝑏〉}:{𝑎}–1-1-onto→{𝑏} |
20 | 8, 16, 19 | vtocl2g 2662 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |