Step | Hyp | Ref
| Expression |
1 | | fococnv2 5095 |
. . . 4
⊢ (𝐹:A–onto→B →
(𝐹 ∘ ◡𝐹) = ( I ↾ B)) |
2 | | cnveq 4452 |
. . . . . 6
⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺) |
3 | 2 | coeq2d 4441 |
. . . . 5
⊢ (𝐹 = 𝐺 → (𝐹 ∘ ◡𝐹) = (𝐹 ∘ ◡𝐺)) |
4 | 3 | eqeq1d 2045 |
. . . 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 5051 |
. . . . 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 5051 |
. . . . 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 5240 |
. . . . . . . . . . . 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 5135 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐺 ∧ x ∈ dom 𝐺) → (𝐺‘x) ∈
V) |
17 | 16 | funfni 4942 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 Fn A ∧ x ∈ A) → (𝐺‘x) ∈
V) |
18 | | vex 2554 |
. . . . . . . . . . . . . 14
⊢ x ∈
V |
19 | | brcnvg 4459 |
. . . . . . . . . . . . . 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 3756 |
. . . . . . . . . . . . 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 5240 |
. . . . . . . . . . . 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 3756 |
. . . . . . . . . . 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 3759 |
. . . . . . . . . . . 12
⊢ (y = x →
((𝐺‘x)◡𝐺y ↔ (𝐺‘x)◡𝐺x)) |
31 | | breq1 3758 |
. . . . . . . . . . . 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 2641 |
. . . . . . . . . 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 5135 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈
V) |
39 | 38 | funfni 4942 |
. . . . . . . . . . 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 4445 |
. . . . . . . . . 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 3757 |
. . . . . . . 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 5049 |
. . . . . . . . . 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 5245 |
. . . . . . . 8
⊢ (((𝐹:A–onto→B ∧ 𝐺:A–onto→B) ∧ x ∈ A) →
(𝐺‘x) ∈ B) |
51 | | fof 5049 |
. . . . . . . . . 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 5245 |
. . . . . . . 8
⊢ (((𝐹:A–onto→B ∧ 𝐺:A–onto→B) ∧ x ∈ A) →
(𝐹‘x) ∈ B) |
54 | | resieq 4565 |
. . . . . . . 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 2042 |
. . . 4
⊢ ((((𝐹:A–onto→B ∧ 𝐺:A–onto→B) ∧ (𝐹 ∘ ◡𝐺) = ( I ↾ B)) ∧ x ∈ A) → (𝐹‘x) = (𝐺‘x)) |
59 | 8, 10, 58 | eqfnfvd 5211 |
. . 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))) |