| Step | Hyp | Ref
| Expression |
| 1 | | ra4v 3885 |
. . . . . . . . . . 11
⊢
(∀𝑤 ∈
Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤))) |
| 2 | | r19.26 3111 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) |
| 3 | 2 | anbi2i 623 |
. . . . . . . . . . . . 13
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))) |
| 4 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) |
| 5 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧) |
| 6 | | predeq3 6325 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧)) |
| 7 | 6 | reseq2d 5997 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) |
| 8 | 5, 7 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
| 9 | 4, 8 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
| 10 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧)) |
| 11 | 6 | reseq2d 5997 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) |
| 12 | 5, 11 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
| 13 | 10, 12 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
| 14 | 9, 13 | anbi12d 632 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑧 → (((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))) |
| 15 | 14 | rspcva 3620 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
| 16 | | eqtr3 2763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹‘𝑧)) |
| 17 | 16 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
| 18 | | eqtr3 2763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧)) |
| 19 | 18 | ex 412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 20 | 17, 19 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 21 | 20 | expimpd 453 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) → (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 22 | | predss 6329 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴 |
| 23 | | fvreseq 7060 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤))) |
| 24 | 22, 23 | mpan2 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤))) |
| 25 | 24 | biimpar 477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) |
| 26 | 25 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
| 27 | 26 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
| 28 | 21, 27 | syl11 33 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 29 | 28 | expd 415 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹‘𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 30 | 15, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 31 | 30 | ex 412 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))) |
| 32 | 31 | com23 86 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))) |
| 33 | 32 | impd 410 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 34 | 3, 33 | biimtrrid 243 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 35 | 34 | a2d 29 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝐴 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 36 | 1, 35 | syl5 34 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 37 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤)) |
| 38 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤)) |
| 39 | 37, 38 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑤) = (𝐺‘𝑤))) |
| 40 | 39 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
| 41 | 36, 40 | frins2 9794 |
. . . . . . . . 9
⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 42 | | rsp 3247 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) → (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 43 | 41, 42 | syl 17 |
. . . . . . . 8
⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 44 | 43 | com3r 87 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 45 | 44 | an4s 660 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 46 | 45 | com12 32 |
. . . . 5
⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 47 | 46 | 3impib 1117 |
. . . 4
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 48 | 47 | ralrimiv 3145 |
. . 3
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)) |
| 49 | | eqid 2737 |
. . 3
⊢ 𝐴 = 𝐴 |
| 50 | 48, 49 | jctil 519 |
. 2
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))) |
| 51 | | eqfnfv2 7052 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 52 | 51 | ad2ant2r 747 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 53 | 52 | 3adant1 1131 |
. 2
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))) |
| 54 | 50, 53 | mpbird 257 |
1
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺) |