| Step | Hyp | Ref
| Expression |
| 1 | | f1cocnv1 6878 |
. . . 4
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
| 2 | | coeq2 5869 |
. . . . 5
⊢ (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ 𝐺)) |
| 3 | 2 | eqeq1d 2739 |
. . . 4
⊢ (𝐹 = 𝐺 → ((◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴) ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
| 4 | 1, 3 | syl5ibcom 245 |
. . 3
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
| 5 | 4 | adantr 480 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
| 6 | | f1fn 6805 |
. . . . . . 7
⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴) |
| 7 | 6 | adantl 481 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴) |
| 8 | 7 | adantr 480 |
. . . . 5
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴) |
| 9 | | f1fn 6805 |
. . . . . . 7
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) |
| 10 | 9 | adantr 480 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴) |
| 11 | 10 | adantr 480 |
. . . . 5
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴) |
| 12 | | equid 2011 |
. . . . . . . . . 10
⊢ 𝑥 = 𝑥 |
| 13 | | resieq 6008 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥( I ↾ 𝐴)𝑥 ↔ 𝑥 = 𝑥)) |
| 14 | 12, 13 | mpbiri 258 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥) |
| 15 | 14 | anidms 566 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥) |
| 16 | 15 | adantl 481 |
. . . . . . 7
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥) |
| 17 | | breq 5145 |
. . . . . . . 8
⊢ ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥)) |
| 18 | 17 | ad2antlr 727 |
. . . . . . 7
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ 𝑥( I ↾ 𝐴)𝑥)) |
| 19 | 16, 18 | mpbird 257 |
. . . . . 6
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥(◡𝐹 ∘ 𝐺)𝑥) |
| 20 | | fnfun 6668 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 Fn 𝐴 → Fun 𝐺) |
| 21 | 7, 20 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐺) |
| 22 | 7 | fndmd 6673 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴) |
| 23 | 22 | eleq2d 2827 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴)) |
| 24 | 23 | biimpar 477 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺) |
| 25 | | funopfvb 6963 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺)) |
| 26 | 21, 24, 25 | syl2an2r 685 |
. . . . . . . . . . . . 13
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺)) |
| 27 | 26 | bicomd 223 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (〈𝑥, 𝑦〉 ∈ 𝐺 ↔ (𝐺‘𝑥) = 𝑦)) |
| 28 | | df-br 5144 |
. . . . . . . . . . . 12
⊢ (𝑥𝐺𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺) |
| 29 | | eqcom 2744 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = 𝑦) |
| 30 | 27, 28, 29 | 3bitr4g 314 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 ↔ 𝑦 = (𝐺‘𝑥))) |
| 31 | 30 | biimpd 229 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 → 𝑦 = (𝐺‘𝑥))) |
| 32 | | df-br 5144 |
. . . . . . . . . . . . 13
⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) |
| 33 | | fnfun 6668 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
| 34 | 10, 33 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐹) |
| 35 | 10 | fndmd 6673 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴) |
| 36 | 35 | eleq2d 2827 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
| 37 | 36 | biimpar 477 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
| 38 | | funopfvb 6963 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
| 39 | 34, 37, 38 | syl2an2r 685 |
. . . . . . . . . . . . 13
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
| 40 | 32, 39 | bitr4id 290 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 ↔ (𝐹‘𝑥) = 𝑦)) |
| 41 | | vex 3484 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
| 42 | | vex 3484 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
| 43 | 41, 42 | brcnv 5893 |
. . . . . . . . . . . 12
⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦) |
| 44 | | eqcom 2744 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) |
| 45 | 40, 43, 44 | 3bitr4g 314 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 ↔ 𝑦 = (𝐹‘𝑥))) |
| 46 | 45 | biimpd 229 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 → 𝑦 = (𝐹‘𝑥))) |
| 47 | 31, 46 | anim12d 609 |
. . . . . . . . 9
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → (𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))) |
| 48 | 47 | eximdv 1917 |
. . . . . . . 8
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))) |
| 49 | 42, 42 | brco 5881 |
. . . . . . . 8
⊢ (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥)) |
| 50 | | fvex 6919 |
. . . . . . . . 9
⊢ (𝐺‘𝑥) ∈ V |
| 51 | 50 | eqvinc 3649 |
. . . . . . . 8
⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))) |
| 52 | 48, 49, 51 | 3imtr4g 296 |
. . . . . . 7
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥))) |
| 53 | 52 | adantlr 715 |
. . . . . 6
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥))) |
| 54 | 19, 53 | mpd 15 |
. . . . 5
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥)) |
| 55 | 8, 11, 54 | eqfnfvd 7054 |
. . . 4
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹) |
| 56 | 55 | eqcomd 2743 |
. . 3
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺) |
| 57 | 56 | ex 412 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺)) |
| 58 | 5, 57 | impbid 212 |
1
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |