Step | Hyp | Ref
| Expression |
1 | | fococnv2 5152 |
. . . 4
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
2 | | cnveq 4509 |
. . . . . 6
⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺) |
3 | 2 | coeq2d 4498 |
. . . . 5
⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺)) |
4 | 3 | eqeq1d 2048 |
. . . 4
⊢ (𝐹 = 𝐺 → ((𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵) ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))) |
5 | 1, 4 | syl5ibcom 144 |
. . 3
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))) |
6 | 5 | adantr 261 |
. 2
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))) |
7 | | fofn 5108 |
. . . . 5
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) |
8 | 7 | ad2antrr 457 |
. . . 4
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴) |
9 | | fofn 5108 |
. . . . 5
⊢ (𝐺:𝐴–onto→𝐵 → 𝐺 Fn 𝐴) |
10 | 9 | ad2antlr 458 |
. . . 4
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴) |
11 | 9 | adantl 262 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺 Fn 𝐴) |
12 | | fnopfv 5297 |
. . . . . . . . . . . 12
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐺‘𝑥)〉 ∈ 𝐺) |
13 | 11, 12 | sylan 267 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐺‘𝑥)〉 ∈ 𝐺) |
14 | 9 | anim1i 323 |
. . . . . . . . . . . . 13
⊢ ((𝐺:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴)) |
15 | 14 | adantll 445 |
. . . . . . . . . . . 12
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴)) |
16 | | funfvex 5192 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐺 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ V) |
17 | 16 | funfni 4999 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ V) |
18 | | vex 2560 |
. . . . . . . . . . . . . 14
⊢ 𝑥 ∈ V |
19 | | brcnvg 4516 |
. . . . . . . . . . . . . 14
⊢ (((𝐺‘𝑥) ∈ V ∧ 𝑥 ∈ V) → ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))) |
20 | 17, 18, 19 | sylancl 392 |
. . . . . . . . . . . . 13
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)◡𝐺𝑥 ↔ 𝑥𝐺(𝐺‘𝑥))) |
21 | | df-br 3765 |
. . . . . . . . . . . . 13
⊢ (𝑥𝐺(𝐺‘𝑥) ↔ 〈𝑥, (𝐺‘𝑥)〉 ∈ 𝐺) |
22 | 20, 21 | syl6bb 185 |
. . . . . . . . . . . 12
⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)◡𝐺𝑥 ↔ 〈𝑥, (𝐺‘𝑥)〉 ∈ 𝐺)) |
23 | 15, 22 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)◡𝐺𝑥 ↔ 〈𝑥, (𝐺‘𝑥)〉 ∈ 𝐺)) |
24 | 13, 23 | mpbird 156 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)◡𝐺𝑥) |
25 | 7 | adantr 261 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹 Fn 𝐴) |
26 | | fnopfv 5297 |
. . . . . . . . . . . 12
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) |
27 | 25, 26 | sylan 267 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) |
28 | | df-br 3765 |
. . . . . . . . . . 11
⊢ (𝑥𝐹(𝐹‘𝑥) ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) |
29 | 27, 28 | sylibr 137 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥)) |
30 | | breq2 3768 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → ((𝐺‘𝑥)◡𝐺𝑦 ↔ (𝐺‘𝑥)◡𝐺𝑥)) |
31 | | breq1 3767 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → (𝑦𝐹(𝐹‘𝑥) ↔ 𝑥𝐹(𝐹‘𝑥))) |
32 | 30, 31 | anbi12d 442 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)) ↔ ((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)))) |
33 | 18, 32 | spcev 2647 |
. . . . . . . . . 10
⊢ (((𝐺‘𝑥)◡𝐺𝑥 ∧ 𝑥𝐹(𝐹‘𝑥)) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥))) |
34 | 24, 29, 33 | syl2anc 391 |
. . . . . . . . 9
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥))) |
35 | 15, 17 | syl 14 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ V) |
36 | 7 | anim1i 323 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴)) |
37 | 36 | adantlr 446 |
. . . . . . . . . . 11
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴)) |
38 | | funfvex 5192 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) |
39 | 38 | funfni 4999 |
. . . . . . . . . . 11
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V) |
40 | 37, 39 | syl 14 |
. . . . . . . . . 10
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V) |
41 | | brcog 4502 |
. . . . . . . . . 10
⊢ (((𝐺‘𝑥) ∈ V ∧ (𝐹‘𝑥) ∈ V) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))) |
42 | 35, 40, 41 | syl2anc 391 |
. . . . . . . . 9
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ ∃𝑦((𝐺‘𝑥)◡𝐺𝑦 ∧ 𝑦𝐹(𝐹‘𝑥)))) |
43 | 34, 42 | mpbird 156 |
. . . . . . . 8
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥)) |
44 | 43 | adantlr 446 |
. . . . . . 7
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥)) |
45 | | breq 3766 |
. . . . . . . 8
⊢ ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))) |
46 | 45 | ad2antlr 458 |
. . . . . . 7
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)(𝐹 ∘ ◡𝐺)(𝐹‘𝑥) ↔ (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥))) |
47 | 44, 46 | mpbid 135 |
. . . . . 6
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥)) |
48 | | fof 5106 |
. . . . . . . . . 10
⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵) |
49 | 48 | adantl 262 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴⟶𝐵) |
50 | 49 | ffvelrnda 5302 |
. . . . . . . 8
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
51 | | fof 5106 |
. . . . . . . . . 10
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) |
52 | 51 | adantr 261 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → 𝐹:𝐴⟶𝐵) |
53 | 52 | ffvelrnda 5302 |
. . . . . . . 8
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
54 | | resieq 4622 |
. . . . . . . 8
⊢ (((𝐺‘𝑥) ∈ 𝐵 ∧ (𝐹‘𝑥) ∈ 𝐵) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))) |
55 | 50, 53, 54 | syl2anc 391 |
. . . . . . 7
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))) |
56 | 55 | adantlr 446 |
. . . . . 6
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥)( I ↾ 𝐵)(𝐹‘𝑥) ↔ (𝐺‘𝑥) = (𝐹‘𝑥))) |
57 | 47, 56 | mpbid 135 |
. . . . 5
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥)) |
58 | 57 | eqcomd 2045 |
. . . 4
⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
59 | 8, 10, 58 | eqfnfvd 5268 |
. . 3
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺) |
60 | 59 | ex 108 |
. 2
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺)) |
61 | 6, 60 | impbid 120 |
1
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))) |