| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | caofref.1 | . . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| 2 |  | caofinv.5 | . . . . . 6
⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))) | 
| 3 |  | caofinv.4 | . . . . . . . 8
⊢ (𝜑 → 𝑁:𝑆⟶𝑆) | 
| 4 | 3 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑁:𝑆⟶𝑆) | 
| 5 |  | caofref.2 | . . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | 
| 6 | 5 | ffvelcdmda 7104 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝑆) | 
| 7 | 4, 6 | ffvelcdmd 7105 | . . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝑁‘(𝐹‘𝑣)) ∈ 𝑆) | 
| 8 | 2, 7 | fmpt3d 7136 | . . . . 5
⊢ (𝜑 → 𝐺:𝐴⟶𝑆) | 
| 9 | 8 | ffvelcdmda 7104 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) | 
| 10 | 5 | ffvelcdmda 7104 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) | 
| 11 |  | fvex 6919 | . . . . . . 7
⊢ (𝑁‘(𝐹‘𝑣)) ∈ V | 
| 12 |  | eqid 2737 | . . . . . . 7
⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) | 
| 13 | 11, 12 | fnmpti 6711 | . . . . . 6
⊢ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴 | 
| 14 | 2 | fneq1d 6661 | . . . . . 6
⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣))) Fn 𝐴)) | 
| 15 | 13, 14 | mpbiri 258 | . . . . 5
⊢ (𝜑 → 𝐺 Fn 𝐴) | 
| 16 |  | dffn5 6967 | . . . . 5
⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) | 
| 17 | 15, 16 | sylib 218 | . . . 4
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) | 
| 18 | 5 | feqmptd 6977 | . . . 4
⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) | 
| 19 | 1, 9, 10, 17, 18 | offval2 7717 | . . 3
⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) | 
| 20 | 2 | fveq1d 6908 | . . . . . . 7
⊢ (𝜑 → (𝐺‘𝑤) = ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤)) | 
| 21 |  | 2fveq3 6911 | . . . . . . . 8
⊢ (𝑣 = 𝑤 → (𝑁‘(𝐹‘𝑣)) = (𝑁‘(𝐹‘𝑤))) | 
| 22 |  | fvex 6919 | . . . . . . . 8
⊢ (𝑁‘(𝐹‘𝑤)) ∈ V | 
| 23 | 21, 12, 22 | fvmpt 7016 | . . . . . . 7
⊢ (𝑤 ∈ 𝐴 → ((𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))‘𝑤) = (𝑁‘(𝐹‘𝑤))) | 
| 24 | 20, 23 | sylan9eq 2797 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) = (𝑁‘(𝐹‘𝑤))) | 
| 25 | 24 | oveq1d 7446 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤))) | 
| 26 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑥 = (𝐹‘𝑤) → (𝑁‘𝑥) = (𝑁‘(𝐹‘𝑤))) | 
| 27 |  | id 22 | . . . . . . . 8
⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | 
| 28 | 26, 27 | oveq12d 7449 | . . . . . . 7
⊢ (𝑥 = (𝐹‘𝑤) → ((𝑁‘𝑥)𝑅𝑥) = ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤))) | 
| 29 | 28 | eqeq1d 2739 | . . . . . 6
⊢ (𝑥 = (𝐹‘𝑤) → (((𝑁‘𝑥)𝑅𝑥) = 𝐵 ↔ ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵)) | 
| 30 |  | caofinvl.6 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵) | 
| 31 | 30 | ralrimiva 3146 | . . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵) | 
| 32 | 31 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ((𝑁‘𝑥)𝑅𝑥) = 𝐵) | 
| 33 | 29, 32, 10 | rspcdva 3623 | . . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑁‘(𝐹‘𝑤))𝑅(𝐹‘𝑤)) = 𝐵) | 
| 34 | 25, 33 | eqtrd 2777 | . . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐹‘𝑤)) = 𝐵) | 
| 35 | 34 | mpteq2dva 5242 | . . 3
⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ 𝐵)) | 
| 36 | 19, 35 | eqtrd 2777 | . 2
⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ 𝐵)) | 
| 37 |  | fconstmpt 5747 | . 2
⊢ (𝐴 × {𝐵}) = (𝑤 ∈ 𝐴 ↦ 𝐵) | 
| 38 | 36, 37 | eqtr4di 2795 | 1
⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝐴 × {𝐵})) |