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Mirrors > Home > MPE Home > Th. List > brcici | Structured version Visualization version GIF version |
Description: Prove that two objects are isomorphic by an explicit isomorphism. (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) |
cic.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
brcici | ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cic.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
2 | eleq1 2877 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌))) | |
3 | 2 | spcegv 3545 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
4 | 1, 1, 3 | sylc 65 | . 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)) |
5 | cic.i | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
6 | cic.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
7 | cic.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | cic.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | cic.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 5, 6, 7, 8, 9 | cic 17061 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
11 | 4, 10 | mpbird 260 | 1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∃wex 1781 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Catccat 16927 Isociso 17008 ≃𝑐 ccic 17057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-supp 7814 df-inv 17010 df-iso 17011 df-cic 17058 |
This theorem is referenced by: cicref 17063 cicsym 17066 cictr 17067 |
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