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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 2726 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invss 17777 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))) |
8 | relxp 5700 | . . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) | |
9 | relss 5787 | . . 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 2726 | . . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
12 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝐶 ∈ Cat) |
13 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑌 ∈ 𝐵) |
14 | 4 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑋 ∈ 𝐵) |
15 | 1, 2, 3, 4, 5, 11 | isinv 17776 | . . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓))) |
16 | 15 | simplbda 498 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓) |
17 | 16 | adantrr 715 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓) |
18 | 1, 2, 3, 4, 5, 11 | isinv 17776 | . . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓))) |
19 | 18 | simprbda 497 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ) |
20 | 19 | adantrl 714 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ) |
21 | 1, 11, 12, 13, 14, 17, 20 | sectcan 17771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ) |
22 | 21 | ex 411 | . . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
23 | 22 | alrimiv 1923 | . . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
24 | 23 | alrimivv 1924 | . 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)) |
25 | dffun2 6564 | . 2 ⊢ (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))) | |
26 | 10, 24, 25 | sylanbrc 581 | 1 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5153 × cxp 5680 Rel wrel 5687 Fun wfun 6548 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 Hom chom 17277 Catccat 17677 Sectcsect 17760 Invcinv 17761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-cat 17681 df-cid 17682 df-sect 17763 df-inv 17764 |
This theorem is referenced by: inviso1 17782 invf 17784 invco 17787 idinv 17805 ffthiso 17951 fuciso 18000 setciso 18113 catciso 18133 rngciso 20616 ringciso 20650 rngcisoALTV 47654 ringcisoALTV 47688 |
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