| Step | Hyp | Ref
| Expression |
| 1 | | functhinc.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
| 2 | | functhinc.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐷) |
| 3 | | functhinc.c |
. . . . . 6
⊢ 𝐶 = (Base‘𝐸) |
| 4 | | functhinc.h |
. . . . . 6
⊢ 𝐻 = (Hom ‘𝐷) |
| 5 | | functhinc.j |
. . . . . 6
⊢ 𝐽 = (Hom ‘𝐸) |
| 6 | | eqid 2737 |
. . . . . 6
⊢
(Id‘𝐷) =
(Id‘𝐷) |
| 7 | | eqid 2737 |
. . . . . 6
⊢
(Id‘𝐸) =
(Id‘𝐸) |
| 8 | | eqid 2737 |
. . . . . 6
⊢
(comp‘𝐷) =
(comp‘𝐷) |
| 9 | | eqid 2737 |
. . . . . 6
⊢
(comp‘𝐸) =
(comp‘𝐸) |
| 10 | | functhinc.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ Cat) |
| 11 | | functhinc.e |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ ThinCat) |
| 12 | 11 | thinccd 49073 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ Cat) |
| 13 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 | isfunc 17909 |
. . . . 5
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))) |
| 14 | | 3anass 1095 |
. . . . 5
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))) |
| 15 | 13, 14 | bitrdi 287 |
. . . 4
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓))))))) |
| 16 | 1, 15 | mpbirand 707 |
. . 3
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))) |
| 17 | | funcf2lem 48914 |
. . . . 5
⊢ (𝐺 ∈ X𝑐 ∈
(𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹‘𝑣)𝐽(𝐹‘𝑢)))) |
| 18 | | functhinc.k |
. . . . . 6
⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
| 19 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑣 ∈ 𝐵) |
| 20 | | simprr 773 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → 𝑢 ∈ 𝐵) |
| 21 | | functhinc.1 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅)) |
| 22 | 21 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅)) |
| 23 | 19, 20, 22 | functhinclem2 49094 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (((𝐹‘𝑣)𝐽(𝐹‘𝑢)) = ∅ → (𝑣𝐻𝑢) = ∅)) |
| 24 | 2, 3, 4, 5, 11, 1,
18, 23 | functhinclem1 49093 |
. . . . 5
⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑣𝐺𝑢):(𝑣𝐻𝑢)⟶((𝐹‘𝑣)𝐽(𝐹‘𝑢))) ↔ 𝐺 = 𝐾)) |
| 25 | 17, 24 | bitrid 283 |
. . . 4
⊢ (𝜑 → (𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ↔ 𝐺 = 𝐾)) |
| 26 | 25 | anbi1d 631 |
. . 3
⊢ (𝜑 → ((𝐺 ∈ X𝑐 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑐))𝐽(𝐹‘(2nd ‘𝑐))) ↑m (𝐻‘𝑐)) ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))) ↔ (𝐺 = 𝐾 ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))) |
| 27 | 16, 26 | bitrd 279 |
. 2
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐺 = 𝐾 ∧ ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))))) |
| 28 | 2, 3, 4, 5, 10, 11, 1, 18, 21, 6, 7, 8,
9 | functhinclem4 49096 |
. 2
⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘((Id‘𝐷)‘𝑎)) = ((Id‘𝐸)‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑓 ∈ (𝑎𝐻𝑏)∀𝑔 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑔(〈𝑎, 𝑏〉(comp‘𝐷)𝑐)𝑓)) = (((𝑏𝐺𝑐)‘𝑔)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉(comp‘𝐸)(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑓)))) |
| 29 | 27, 28 | mpbiran3d 48717 |
1
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾)) |