| Step | Hyp | Ref
| Expression |
| 1 | | f1of 6848 |
. . . . . . . . 9
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶𝐵) |
| 2 | | ssun1 4178 |
. . . . . . . . . 10
⊢ 𝐵 ⊆ (𝐵 ∪ {𝑌}) |
| 3 | 2 | a1i 11 |
. . . . . . . . 9
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐵 ⊆ (𝐵 ∪ {𝑌})) |
| 4 | 1, 3 | fssd 6753 |
. . . . . . . 8
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶(𝐵 ∪ {𝑌})) |
| 5 | 4 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝐺:𝐴⟶(𝐵 ∪ {𝑌})) |
| 6 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑋 ∈ 𝑉) |
| 7 | | df-nel 3047 |
. . . . . . . . . . 11
⊢ (𝑋 ∉ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴) |
| 8 | 7 | biimpi 216 |
. . . . . . . . . 10
⊢ (𝑋 ∉ 𝐴 → ¬ 𝑋 ∈ 𝐴) |
| 9 | 8 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵) → ¬ 𝑋 ∈ 𝐴) |
| 10 | 6, 9 | anim12i 613 |
. . . . . . . 8
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴)) |
| 11 | 10 | 3adant1 1131 |
. . . . . . 7
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴)) |
| 12 | | eqid 2737 |
. . . . . . . . . . 11
⊢ 𝑌 = 𝑌 |
| 13 | 12 | olci 867 |
. . . . . . . . . 10
⊢ (𝑌 ∈ 𝐵 ∨ 𝑌 = 𝑌) |
| 14 | | elunsn 4683 |
. . . . . . . . . 10
⊢ (𝑌 ∈ 𝑊 → (𝑌 ∈ (𝐵 ∪ {𝑌}) ↔ (𝑌 ∈ 𝐵 ∨ 𝑌 = 𝑌))) |
| 15 | 13, 14 | mpbiri 258 |
. . . . . . . . 9
⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ (𝐵 ∪ {𝑌})) |
| 16 | 15 | adantl 481 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑌 ∈ (𝐵 ∪ {𝑌})) |
| 17 | 16 | 3ad2ant2 1135 |
. . . . . . 7
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝑌 ∈ (𝐵 ∪ {𝑌})) |
| 18 | 5, 11, 17 | 3jca 1129 |
. . . . . 6
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝐺:𝐴⟶(𝐵 ∪ {𝑌}) ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑌 ∈ (𝐵 ∪ {𝑌}))) |
| 19 | | fsnunf 7205 |
. . . . . 6
⊢ ((𝐺:𝐴⟶(𝐵 ∪ {𝑌}) ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑌 ∈ (𝐵 ∪ {𝑌})) → (𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌})) |
| 20 | 18, 19 | syl 17 |
. . . . 5
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌})) |
| 21 | | f1of1 6847 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴–1-1→𝐵) |
| 22 | | dff14a 7290 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐺‘𝑎) ≠ (𝐺‘𝑏)))) |
| 23 | | neeq1 3003 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝑥 → (𝑎 ≠ 𝑏 ↔ 𝑥 ≠ 𝑏)) |
| 24 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = 𝑥 → (𝐺‘𝑎) = (𝐺‘𝑥)) |
| 25 | 24 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝑥 → ((𝐺‘𝑎) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑥) ≠ (𝐺‘𝑏))) |
| 26 | 23, 25 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑥 → ((𝑎 ≠ 𝑏 → (𝐺‘𝑎) ≠ (𝐺‘𝑏)) ↔ (𝑥 ≠ 𝑏 → (𝐺‘𝑥) ≠ (𝐺‘𝑏)))) |
| 27 | | neeq2 3004 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑦 → (𝑥 ≠ 𝑏 ↔ 𝑥 ≠ 𝑦)) |
| 28 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 = 𝑦 → (𝐺‘𝑏) = (𝐺‘𝑦)) |
| 29 | 28 | neeq2d 3001 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑦 → ((𝐺‘𝑥) ≠ (𝐺‘𝑏) ↔ (𝐺‘𝑥) ≠ (𝐺‘𝑦))) |
| 30 | 27, 29 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑦 → ((𝑥 ≠ 𝑏 → (𝐺‘𝑥) ≠ (𝐺‘𝑏)) ↔ (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 31 | 26, 30 | rspc2va 3634 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐺‘𝑎) ≠ (𝐺‘𝑏))) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))) |
| 32 | 31 | expcom 413 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑎 ∈
𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐺‘𝑎) ≠ (𝐺‘𝑏)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 33 | 32 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎 ≠ 𝑏 → (𝐺‘𝑎) ≠ (𝐺‘𝑏))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 34 | 22, 33 | sylbi 217 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:𝐴–1-1→𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 35 | 21, 34 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝐺:𝐴–1-1-onto→𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 36 | 35 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦)))) |
| 37 | 36 | impl 455 |
. . . . . . . . . . . 12
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → (𝐺‘𝑥) ≠ (𝐺‘𝑦))) |
| 38 | 37 | imp 406 |
. . . . . . . . . . 11
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → (𝐺‘𝑥) ≠ (𝐺‘𝑦)) |
| 39 | | nelne2 3040 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴) → 𝑥 ≠ 𝑋) |
| 40 | 39 | necomd 2996 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴) → 𝑋 ≠ 𝑥) |
| 41 | 40 | expcom 413 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑋 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥)) |
| 42 | 7, 41 | sylbi 217 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∉ 𝐴 → (𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥)) |
| 43 | 42 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵) → (𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥)) |
| 44 | 43 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝑥 ∈ 𝐴 → 𝑋 ≠ 𝑥)) |
| 45 | 44 | imp 406 |
. . . . . . . . . . . . . 14
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑋 ≠ 𝑥) |
| 46 | 45 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑋 ≠ 𝑥) |
| 47 | 46 | adantr 480 |
. . . . . . . . . . . 12
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → 𝑋 ≠ 𝑥) |
| 48 | | fvunsn 7199 |
. . . . . . . . . . . 12
⊢ (𝑋 ≠ 𝑥 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) = (𝐺‘𝑥)) |
| 49 | 47, 48 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) = (𝐺‘𝑥)) |
| 50 | | nelne2 3040 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴) → 𝑦 ≠ 𝑋) |
| 51 | 50 | necomd 2996 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴) → 𝑋 ≠ 𝑦) |
| 52 | 51 | expcom 413 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑋 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦)) |
| 53 | 7, 52 | sylbi 217 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∉ 𝐴 → (𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦)) |
| 54 | 53 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵) → (𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦)) |
| 55 | 54 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦)) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 → 𝑋 ≠ 𝑦)) |
| 57 | 56 | imp 406 |
. . . . . . . . . . . . 13
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑋 ≠ 𝑦) |
| 58 | 57 | adantr 480 |
. . . . . . . . . . . 12
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → 𝑋 ≠ 𝑦) |
| 59 | | fvunsn 7199 |
. . . . . . . . . . . 12
⊢ (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) = (𝐺‘𝑦)) |
| 60 | 58, 59 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) = (𝐺‘𝑦)) |
| 61 | 38, 49, 60 | 3netr4d 3018 |
. . . . . . . . . 10
⊢
(((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) |
| 62 | 61 | ex 412 |
. . . . . . . . 9
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 63 | 62 | ralrimiva 3146 |
. . . . . . . 8
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 64 | 1 | 3ad2ant1 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝐺:𝐴⟶𝐵) |
| 65 | 64 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
| 66 | | df-nel 3047 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑌 ∉ 𝐵 ↔ ¬ 𝑌 ∈ 𝐵) |
| 67 | 66 | biimpi 216 |
. . . . . . . . . . . . . . . 16
⊢ (𝑌 ∉ 𝐵 → ¬ 𝑌 ∈ 𝐵) |
| 68 | 67 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵) → ¬ 𝑌 ∈ 𝐵) |
| 69 | 68 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ¬ 𝑌 ∈ 𝐵) |
| 70 | 69 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑌 ∈ 𝐵) |
| 71 | | nelne2 3040 |
. . . . . . . . . . . . 13
⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵) → (𝐺‘𝑥) ≠ 𝑌) |
| 72 | 65, 70, 71 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ≠ 𝑌) |
| 73 | 72 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → (𝐺‘𝑥) ≠ 𝑌) |
| 74 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → 𝑥 ≠ 𝑋) |
| 75 | 74 | necomd 2996 |
. . . . . . . . . . . 12
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → 𝑋 ≠ 𝑥) |
| 76 | 75, 48 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) = (𝐺‘𝑥)) |
| 77 | 6 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝑋 ∈ 𝑉) |
| 78 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝑌 ∈ 𝑊) |
| 79 | 78 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝑌 ∈ 𝑊) |
| 80 | | f1odm 6852 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐺:𝐴–1-1-onto→𝐵 → dom 𝐺 = 𝐴) |
| 81 | 80 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐴 = dom 𝐺) |
| 82 | 81 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐺:𝐴–1-1-onto→𝐵 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺)) |
| 83 | 82 | notbid 318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐺:𝐴–1-1-onto→𝐵 → (¬ 𝑋 ∈ 𝐴 ↔ ¬ 𝑋 ∈ dom 𝐺)) |
| 84 | 7, 83 | bitrid 283 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺:𝐴–1-1-onto→𝐵 → (𝑋 ∉ 𝐴 ↔ ¬ 𝑋 ∈ dom 𝐺)) |
| 85 | 84 | biimpd 229 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺:𝐴–1-1-onto→𝐵 → (𝑋 ∉ 𝐴 → ¬ 𝑋 ∈ dom 𝐺)) |
| 86 | 85 | adantrd 491 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:𝐴–1-1-onto→𝐵 → ((𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵) → ¬ 𝑋 ∈ dom 𝐺)) |
| 87 | 86 | imp 406 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ¬ 𝑋 ∈ dom 𝐺) |
| 88 | 87 | 3adant2 1132 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ¬ 𝑋 ∈ dom 𝐺) |
| 89 | 77, 79, 88 | 3jca 1129 |
. . . . . . . . . . . . . 14
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺)) |
| 90 | 89 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺)) |
| 91 | 90 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺)) |
| 92 | | fsnunfv 7207 |
. . . . . . . . . . . 12
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
| 93 | 91, 92 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
| 94 | 73, 76, 93 | 3netr4d 3018 |
. . . . . . . . . 10
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≠ 𝑋) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
| 95 | 94 | ex 412 |
. . . . . . . . 9
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋))) |
| 96 | 77 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ 𝑉) |
| 97 | | neeq2 3004 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑋 → (𝑥 ≠ 𝑦 ↔ 𝑥 ≠ 𝑋)) |
| 98 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) = ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
| 99 | 98 | neeq2d 3001 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑋 → (((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) ↔ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋))) |
| 100 | 97, 99 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑋 → ((𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑥 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 101 | 100 | ralsng 4675 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝑉 → (∀𝑦 ∈ {𝑋} (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑥 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 102 | 96, 101 | syl 17 |
. . . . . . . . 9
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ {𝑋} (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑥 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 103 | 95, 102 | mpbird 257 |
. . . . . . . 8
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ {𝑋} (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 104 | | ralun 4198 |
. . . . . . . 8
⊢
((∀𝑦 ∈
𝐴 (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ∧ ∀𝑦 ∈ {𝑋} (𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) → ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 105 | 63, 103, 104 | syl2anc 584 |
. . . . . . 7
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 106 | 105 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 107 | 64 | ffvelcdmda 7104 |
. . . . . . . . . . . . . 14
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ 𝐵) |
| 108 | 69 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑌 ∈ 𝐵) |
| 109 | 107, 108 | jca 511 |
. . . . . . . . . . . . 13
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵)) |
| 110 | 109 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → ((𝐺‘𝑦) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵)) |
| 111 | | nelne2 3040 |
. . . . . . . . . . . . 13
⊢ (((𝐺‘𝑦) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵) → (𝐺‘𝑦) ≠ 𝑌) |
| 112 | 111 | necomd 2996 |
. . . . . . . . . . . 12
⊢ (((𝐺‘𝑦) ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐵) → 𝑌 ≠ (𝐺‘𝑦)) |
| 113 | 110, 112 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → 𝑌 ≠ (𝐺‘𝑦)) |
| 114 | 89 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺)) |
| 115 | 114 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐺)) |
| 116 | 115, 92 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
| 117 | 59 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) = (𝐺‘𝑦)) |
| 118 | 113, 116,
117 | 3netr4d 3018 |
. . . . . . . . . 10
⊢ ((((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ≠ 𝑦) → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) |
| 119 | 118 | ex 412 |
. . . . . . . . 9
⊢ (((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) ∧ 𝑦 ∈ 𝐴) → (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 120 | 119 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑦 ∈ 𝐴 (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 121 | | eqid 2737 |
. . . . . . . . . 10
⊢ 𝑋 = 𝑋 |
| 122 | | eqneqall 2951 |
. . . . . . . . . 10
⊢ (𝑋 = 𝑋 → (𝑋 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋))) |
| 123 | 121, 122 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑋 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
| 124 | | neeq2 3004 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑋 → (𝑋 ≠ 𝑦 ↔ 𝑋 ≠ 𝑋)) |
| 125 | 98 | neeq2d 3001 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑋 → (((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) ↔ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋))) |
| 126 | 124, 125 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑋 → ((𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑋 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 127 | 126 | ralsng 4675 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝑉 → (∀𝑦 ∈ {𝑋} (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑋 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 128 | 77, 127 | syl 17 |
. . . . . . . . 9
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (∀𝑦 ∈ {𝑋} (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑋 ≠ 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)))) |
| 129 | 123, 128 | mpbiri 258 |
. . . . . . . 8
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑦 ∈ {𝑋} (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 130 | | ralun 4198 |
. . . . . . . 8
⊢
((∀𝑦 ∈
𝐴 (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ∧ ∀𝑦 ∈ {𝑋} (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) → ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 131 | 120, 129,
130 | syl2anc 584 |
. . . . . . 7
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 132 | | neeq1 3003 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑥 ≠ 𝑦 ↔ 𝑋 ≠ 𝑦)) |
| 133 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) = ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
| 134 | 133 | neeq1d 3000 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦) ↔ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 135 | 132, 134 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ (𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 136 | 135 | ralbidv 3178 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 137 | 136 | ralsng 4675 |
. . . . . . . 8
⊢ (𝑋 ∈ 𝑉 → (∀𝑥 ∈ {𝑋}∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 138 | 77, 137 | syl 17 |
. . . . . . 7
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (∀𝑥 ∈ {𝑋}∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑋 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑋) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 139 | 131, 138 | mpbird 257 |
. . . . . 6
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑥 ∈ {𝑋}∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 140 | | ralun 4198 |
. . . . . 6
⊢
((∀𝑥 ∈
𝐴 ∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)) ∧ ∀𝑥 ∈ {𝑋}∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) → ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 141 | 106, 139,
140 | syl2anc 584 |
. . . . 5
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) |
| 142 | 20, 141 | jca 511 |
. . . 4
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌}) ∧ ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 143 | | rnun 6165 |
. . . . 5
⊢ ran
(𝐺 ∪ {〈𝑋, 𝑌〉}) = (ran 𝐺 ∪ ran {〈𝑋, 𝑌〉}) |
| 144 | | f1ofo 6855 |
. . . . . . . 8
⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴–onto→𝐵) |
| 145 | | forn 6823 |
. . . . . . . 8
⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵) |
| 146 | 144, 145 | syl 17 |
. . . . . . 7
⊢ (𝐺:𝐴–1-1-onto→𝐵 → ran 𝐺 = 𝐵) |
| 147 | 146 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ran 𝐺 = 𝐵) |
| 148 | | rnsnopg 6241 |
. . . . . . . 8
⊢ (𝑋 ∈ 𝑉 → ran {〈𝑋, 𝑌〉} = {𝑌}) |
| 149 | 148 | adantr 480 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ran {〈𝑋, 𝑌〉} = {𝑌}) |
| 150 | 149 | 3ad2ant2 1135 |
. . . . . 6
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ran {〈𝑋, 𝑌〉} = {𝑌}) |
| 151 | 147, 150 | uneq12d 4169 |
. . . . 5
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (ran 𝐺 ∪ ran {〈𝑋, 𝑌〉}) = (𝐵 ∪ {𝑌})) |
| 152 | 143, 151 | eqtrid 2789 |
. . . 4
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → ran (𝐺 ∪ {〈𝑋, 𝑌〉}) = (𝐵 ∪ {𝑌})) |
| 153 | 142, 152 | jca 511 |
. . 3
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌}) ∧ ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) ∧ ran (𝐺 ∪ {〈𝑋, 𝑌〉}) = (𝐵 ∪ {𝑌}))) |
| 154 | | dff1o5 6857 |
. . . 4
⊢ ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}) ↔ ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1→(𝐵 ∪ {𝑌}) ∧ ran (𝐺 ∪ {〈𝑋, 𝑌〉}) = (𝐵 ∪ {𝑌}))) |
| 155 | | dff14a 7290 |
. . . 4
⊢ ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1→(𝐵 ∪ {𝑌}) ↔ ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌}) ∧ ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦)))) |
| 156 | 154, 155 | bianbi 627 |
. . 3
⊢ ((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}) ↔ (((𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})⟶(𝐵 ∪ {𝑌}) ∧ ∀𝑥 ∈ (𝐴 ∪ {𝑋})∀𝑦 ∈ (𝐴 ∪ {𝑋})(𝑥 ≠ 𝑦 → ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑥) ≠ ((𝐺 ∪ {〈𝑋, 𝑌〉})‘𝑦))) ∧ ran (𝐺 ∪ {〈𝑋, 𝑌〉}) = (𝐵 ∪ {𝑌}))) |
| 157 | 153, 156 | sylibr 234 |
. 2
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → (𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌})) |
| 158 | | f1ounsn.f |
. . 3
⊢ 𝐹 = (𝐺 ∪ {〈𝑋, 𝑌〉}) |
| 159 | | f1oeq1 6836 |
. . 3
⊢ (𝐹 = (𝐺 ∪ {〈𝑋, 𝑌〉}) → (𝐹:(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}) ↔ (𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}))) |
| 160 | 158, 159 | ax-mp 5 |
. 2
⊢ (𝐹:(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}) ↔ (𝐺 ∪ {〈𝑋, 𝑌〉}):(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌})) |
| 161 | 157, 160 | sylibr 234 |
1
⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝐹:(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌})) |