| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | id 22 | . . . . 5
⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | 
| 2 | 1, 1 | oveq12d 7450 | . . . 4
⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑥) = (𝑋𝐺𝑋)) | 
| 3 |  | fveq2 6905 | . . . 4
⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | 
| 4 | 2, 3 | fveq12d 6912 | . . 3
⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = ((𝑋𝐺𝑋)‘( 1 ‘𝑋))) | 
| 5 |  | 2fveq3 6910 | . . 3
⊢ (𝑥 = 𝑋 → (𝐼‘(𝐹‘𝑥)) = (𝐼‘(𝐹‘𝑋))) | 
| 6 | 4, 5 | eqeq12d 2752 | . 2
⊢ (𝑥 = 𝑋 → (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ↔ ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))) | 
| 7 |  | funcid.f | . . . . 5
⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | 
| 8 |  | funcid.b | . . . . . 6
⊢ 𝐵 = (Base‘𝐷) | 
| 9 |  | eqid 2736 | . . . . . 6
⊢
(Base‘𝐸) =
(Base‘𝐸) | 
| 10 |  | eqid 2736 | . . . . . 6
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) | 
| 11 |  | eqid 2736 | . . . . . 6
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) | 
| 12 |  | funcid.1 | . . . . . 6
⊢  1 =
(Id‘𝐷) | 
| 13 |  | funcid.i | . . . . . 6
⊢ 𝐼 = (Id‘𝐸) | 
| 14 |  | eqid 2736 | . . . . . 6
⊢
(comp‘𝐷) =
(comp‘𝐷) | 
| 15 |  | eqid 2736 | . . . . . 6
⊢
(comp‘𝐸) =
(comp‘𝐸) | 
| 16 |  | df-br 5143 | . . . . . . . . 9
⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | 
| 17 | 7, 16 | sylib 218 | . . . . . . . 8
⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | 
| 18 |  | funcrcl 17909 | . . . . . . . 8
⊢
(〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | 
| 19 | 17, 18 | syl 17 | . . . . . . 7
⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | 
| 20 | 19 | simpld 494 | . . . . . 6
⊢ (𝜑 → 𝐷 ∈ Cat) | 
| 21 | 19 | simprd 495 | . . . . . 6
⊢ (𝜑 → 𝐸 ∈ Cat) | 
| 22 | 8, 9, 10, 11, 12, 13, 14, 15, 20, 21 | isfunc 17910 | . . . . 5
⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom
‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))) | 
| 23 | 7, 22 | mpbid 232 | . . . 4
⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑧))) ↑m ((Hom
‘𝐷)‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))) | 
| 24 | 23 | simp3d 1144 | . . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))) | 
| 25 |  | simpl 482 | . . . 4
⊢ ((((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥))) | 
| 26 | 25 | ralimi 3082 | . . 3
⊢
(∀𝑥 ∈
𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑥𝐺𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))) → ∀𝑥 ∈ 𝐵 ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥))) | 
| 27 | 24, 26 | syl 17 | . 2
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥))) | 
| 28 |  | funcid.x | . 2
⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| 29 | 6, 27, 28 | rspcdva 3622 | 1
⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋))) |