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| Mirrors > Home > MPE Home > Th. List > invfun | Structured version Visualization version GIF version | ||
| Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| invfval.b | ⊢ 𝐵 = (Base‘𝐶) |
| invfval.n | ⊢ 𝑁 = (Inv‘𝐶) |
| invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| invfun | ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | invfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | invfval.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
| 3 | invfval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | invfval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | invfval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 7 | 1, 2, 3, 4, 5, 6 | invss 17805 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))) |
| 8 | relxp 5703 | . . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) | |
| 9 | relss 5791 | . . 3 ⊢ ((𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → (Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → Rel (𝑋𝑁𝑌))) | |
| 10 | 7, 8, 9 | mpisyl 21 | . 2 ⊢ (𝜑 → Rel (𝑋𝑁𝑌)) |
| 11 | eqid 2737 | . . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
| 12 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝐶 ∈ Cat) |
| 13 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑌 ∈ 𝐵) |
| 14 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑋 ∈ 𝐵) |
| 15 | 1, 2, 3, 4, 5, 11 | isinv 17804 | . . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓))) |
| 16 | 15 | simplbda 499 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓) |
| 17 | 16 | adantrr 717 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓) |
| 18 | 1, 2, 3, 4, 5, 11 | isinv 17804 | . . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓))) |
| 19 | 18 | simprbda 498 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ) |
| 20 | 19 | adantrl 716 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ) |
| 21 | 1, 11, 12, 13, 14, 17, 20 | sectcan 17799 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ) |
| 22 | 21 | ex 412 | . . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
| 23 | 22 | alrimiv 1927 | . . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
| 24 | 23 | alrimivv 1928 | . 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
| 25 | dffun2 6571 | . 2 ⊢ (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))) | |
| 26 | 10, 24, 25 | sylanbrc 583 | 1 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 Sectcsect 17788 Invcinv 17789 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cat 17711 df-cid 17712 df-sect 17791 df-inv 17792 |
| This theorem is referenced by: inviso1 17810 invf 17812 invco 17815 idinv 17833 ffthiso 17976 fuciso 18023 setciso 18136 catciso 18156 rngciso 20638 ringciso 20672 rngcisoALTV 48193 ringcisoALTV 48227 |
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