Step | Hyp | Ref
| Expression |
1 | | f1cocnv1 6690 |
. . . 4
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
2 | | coeq2 5727 |
. . . . 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 248 |
. . 3
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
5 | 4 | adantr 484 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
6 | | f1fn 6616 |
. . . . . . 7
⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺 Fn 𝐴) |
7 | 6 | adantl 485 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐺 Fn 𝐴) |
8 | 7 | adantr 484 |
. . . . 5
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴) |
9 | | f1fn 6616 |
. . . . . . 7
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) |
10 | 9 | adantr 484 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴) |
11 | 10 | adantr 484 |
. . . . 5
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴) |
12 | | equid 2020 |
. . . . . . . . . 10
⊢ 𝑥 = 𝑥 |
13 | | resieq 5862 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥( I ↾ 𝐴)𝑥 ↔ 𝑥 = 𝑥)) |
14 | 12, 13 | mpbiri 261 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥) |
15 | 14 | anidms 570 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → 𝑥( I ↾ 𝐴)𝑥) |
16 | 15 | adantl 485 |
. . . . . . 7
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥( I ↾ 𝐴)𝑥) |
17 | | breq 5055 |
. . . . . . . 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 260 |
. . . . . 6
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥(◡𝐹 ∘ 𝐺)𝑥) |
20 | | fnfun 6479 |
. . . . . . . . . . . . . . 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 | | fndm 6481 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴) |
23 | 7, 22 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐺 = 𝐴) |
24 | 23 | eleq2d 2823 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝐴)) |
25 | 24 | biimpar 481 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐺) |
26 | | funopfvb 6768 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐺 ∧ 𝑥 ∈ dom 𝐺) → ((𝐺‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺)) |
27 | 21, 25, 26 | syl2an2r 685 |
. . . . . . . . . . . . 13
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺)) |
28 | 27 | bicomd 226 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (〈𝑥, 𝑦〉 ∈ 𝐺 ↔ (𝐺‘𝑥) = 𝑦)) |
29 | | df-br 5054 |
. . . . . . . . . . . 12
⊢ (𝑥𝐺𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐺) |
30 | | eqcom 2744 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘𝑥) ↔ (𝐺‘𝑥) = 𝑦) |
31 | 28, 29, 30 | 3bitr4g 317 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 ↔ 𝑦 = (𝐺‘𝑥))) |
32 | 31 | biimpd 232 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐺𝑦 → 𝑦 = (𝐺‘𝑥))) |
33 | | df-br 5054 |
. . . . . . . . . . . . 13
⊢ (𝑥𝐹𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹) |
34 | | fnfun 6479 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
35 | 10, 34 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → Fun 𝐹) |
36 | | fndm 6481 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) |
37 | 10, 36 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴) |
38 | 37 | eleq2d 2823 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
39 | 38 | biimpar 481 |
. . . . . . . . . . . . . 14
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
40 | | funopfvb 6768 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
41 | 35, 39, 40 | syl2an2r 685 |
. . . . . . . . . . . . 13
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
42 | 33, 41 | bitr4id 293 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝑦 ↔ (𝐹‘𝑥) = 𝑦)) |
43 | | vex 3412 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
44 | | vex 3412 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
45 | 43, 44 | brcnv 5751 |
. . . . . . . . . . . 12
⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦) |
46 | | eqcom 2744 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) |
47 | 42, 45, 46 | 3bitr4g 317 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 ↔ 𝑦 = (𝐹‘𝑥))) |
48 | 47 | biimpd 232 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝐹𝑥 → 𝑦 = (𝐹‘𝑥))) |
49 | 32, 48 | anim12d 612 |
. . . . . . . . 9
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → (𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))) |
50 | 49 | eximdv 1925 |
. . . . . . . 8
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥) → ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥)))) |
51 | 44, 44 | brco 5739 |
. . . . . . . 8
⊢ (𝑥(◡𝐹 ∘ 𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦 ∧ 𝑦◡𝐹𝑥)) |
52 | | fvex 6730 |
. . . . . . . . 9
⊢ (𝐺‘𝑥) ∈ V |
53 | 52 | eqvinc 3556 |
. . . . . . . 8
⊢ ((𝐺‘𝑥) = (𝐹‘𝑥) ↔ ∃𝑦(𝑦 = (𝐺‘𝑥) ∧ 𝑦 = (𝐹‘𝑥))) |
54 | 50, 51, 53 | 3imtr4g 299 |
. . . . . . 7
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(◡𝐹 ∘ 𝐺)𝑥 → (𝐺‘𝑥) = (𝐹‘𝑥))) |
55 | 54 | 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))) |
56 | 19, 55 | mpd 15 |
. . . . 5
⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥)) |
57 | 8, 11, 56 | eqfnfvd 6855 |
. . . 4
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹) |
58 | 57 | eqcomd 2743 |
. . 3
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺) |
59 | 58 | ex 416 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → ((◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺)) |
60 | 5, 59 | impbid 215 |
1
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |