|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > cic | Structured version Visualization version GIF version | ||
| Description: Objects 𝑋 and 𝑌 in a category are isomorphic provided that there is an isomorphism 𝑓:𝑋⟶𝑌, see definition 3.15 of [Adamek] p. 29. (Contributed by AV, 4-Apr-2020.) | 
| Ref | Expression | 
|---|---|
| cic.i | ⊢ 𝐼 = (Iso‘𝐶) | 
| cic.b | ⊢ 𝐵 = (Base‘𝐶) | 
| cic.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| cic.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| cic.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| cic | ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cic.i | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
| 2 | cic.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 3 | cic.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | cic.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | cic.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | 1, 2, 3, 4, 5 | brcic 17842 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅)) | 
| 7 | n0 4353 | . 2 ⊢ ((𝑋𝐼𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)) | |
| 8 | 6, 7 | bitrdi 287 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Catccat 17707 Isociso 17790 ≃𝑐 ccic 17839 | 
| 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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-supp 8186 df-inv 17792 df-iso 17793 df-cic 17840 | 
| This theorem is referenced by: brcici 17844 cicsym 17848 cictr 17849 initoeu1w 18057 initoeu2 18061 termoeu1w 18064 nzerooringczr 21491 thincciso 49102 thincciso4 49106 thinccic 49118 | 
| Copyright terms: Public domain | W3C validator |