| Step | Hyp | Ref
| Expression |
| 1 | | f1ocnv 6860 |
. . . . 5
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) |
| 2 | 1 | adantr 480 |
. . . 4
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ◡𝐹:𝑌–1-1-onto→𝑋) |
| 3 | | f1ocnvdm 7305 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (◡𝐹‘𝑢) ∈ 𝑋) |
| 4 | 3 | ex 412 |
. . . . . . . . . 10
⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑢 ∈ 𝑌 → (◡𝐹‘𝑢) ∈ 𝑋)) |
| 5 | | f1ocnvdm 7305 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (◡𝐹‘𝑣) ∈ 𝑋) |
| 6 | 5 | ex 412 |
. . . . . . . . . 10
⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑣 ∈ 𝑌 → (◡𝐹‘𝑣) ∈ 𝑋)) |
| 7 | 4, 6 | anim12d 609 |
. . . . . . . . 9
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋))) |
| 8 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋))) |
| 9 | 8 | imdistani 568 |
. . . . . . 7
⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋))) |
| 10 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑥 = (◡𝐹‘𝑢) → (𝑥𝑀𝑦) = ((◡𝐹‘𝑢)𝑀𝑦)) |
| 11 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = (◡𝐹‘𝑢) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑢))) |
| 12 | 11 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦))) |
| 13 | 10, 12 | eqeq12d 2753 |
. . . . . . . . . 10
⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)))) |
| 14 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑦 = (◡𝐹‘𝑣) → ((◡𝐹‘𝑢)𝑀𝑦) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))) |
| 15 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑦 = (◡𝐹‘𝑣) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑣))) |
| 16 | 15 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑦 = (◡𝐹‘𝑣) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))) |
| 17 | 14, 16 | eqeq12d 2753 |
. . . . . . . . . 10
⊢ (𝑦 = (◡𝐹‘𝑣) → (((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))) |
| 18 | 13, 17 | rspc2v 3633 |
. . . . . . . . 9
⊢ (((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))) |
| 19 | 18 | impcom 407 |
. . . . . . . 8
⊢
((∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))) |
| 20 | 19 | adantll 714 |
. . . . . . 7
⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))) |
| 21 | 9, 20 | syl 17 |
. . . . . 6
⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))) |
| 22 | | f1ocnvfv2 7297 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢) |
| 23 | 22 | adantrr 717 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢) |
| 24 | | f1ocnvfv2 7297 |
. . . . . . . . 9
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣) |
| 25 | 24 | adantrl 716 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣) |
| 26 | 23, 25 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣)) |
| 27 | 26 | adantlr 715 |
. . . . . 6
⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣)) |
| 28 | 21, 27 | eqtr2d 2778 |
. . . . 5
⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))) |
| 29 | 28 | ralrimivva 3202 |
. . . 4
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))) |
| 30 | 2, 29 | jca 511 |
. . 3
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))) |
| 31 | 30 | a1i 11 |
. 2
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))))) |
| 32 | | isismty 37808 |
. 2
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))) |
| 33 | | isismty 37808 |
. . 3
⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))))) |
| 34 | 33 | ancoms 458 |
. 2
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))))) |
| 35 | 31, 32, 34 | 3imtr4d 294 |
1
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀))) |