Proof of Theorem funcf2lem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elixp2 8942 | . 2
⊢ (𝐺 ∈ X𝑧 ∈
(𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))) | 
| 2 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐺‘𝑧) = (𝐺‘〈𝑥, 𝑦〉)) | 
| 3 |  | df-ov 7435 | . . . . . . 7
⊢ (𝑥𝐺𝑦) = (𝐺‘〈𝑥, 𝑦〉) | 
| 4 | 2, 3 | eqtr4di 2794 | . . . . . 6
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐺‘𝑧) = (𝑥𝐺𝑦)) | 
| 5 |  | vex 3483 | . . . . . . . . . 10
⊢ 𝑥 ∈ V | 
| 6 |  | vex 3483 | . . . . . . . . . 10
⊢ 𝑦 ∈ V | 
| 7 | 5, 6 | op1std 8025 | . . . . . . . . 9
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) | 
| 8 | 7 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥)) | 
| 9 | 5, 6 | op2ndd 8026 | . . . . . . . . 9
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) | 
| 10 | 9 | fveq2d 6909 | . . . . . . . 8
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦)) | 
| 11 | 8, 10 | oveq12d 7450 | . . . . . . 7
⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) | 
| 12 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | 
| 13 |  | df-ov 7435 | . . . . . . . 8
⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | 
| 14 | 12, 13 | eqtr4di 2794 | . . . . . . 7
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) | 
| 15 | 11, 14 | oveq12d 7450 | . . . . . 6
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦))) | 
| 16 | 4, 15 | eleq12d 2834 | . . . . 5
⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))) | 
| 17 |  | ovex 7465 | . . . . . 6
⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V | 
| 18 |  | ovex 7465 | . . . . . 6
⊢ (𝑥𝐻𝑦) ∈ V | 
| 19 | 17, 18 | elmap 8912 | . . . . 5
⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))) | 
| 20 | 16, 19 | bitrdi 287 | . . . 4
⊢ (𝑧 = 〈𝑥, 𝑦〉 → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) | 
| 21 | 20 | ralxp 5851 | . . 3
⊢
(∀𝑧 ∈
(𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))) | 
| 22 | 21 | 3anbi3i 1159 | . 2
⊢ ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) | 
| 23 | 1, 22 | bitri 275 | 1
⊢ (𝐺 ∈ X𝑧 ∈
(𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |