Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2902 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋𝐼𝑌))) | |
3 | 2 | spcegv 3599 | . . 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 17071 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌))) |
11 | 4, 10 | mpbird 259 | 1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Catccat 16937 Isociso 17018 ≃𝑐 ccic 17067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-supp 7833 df-inv 17020 df-iso 17021 df-cic 17068 |
This theorem is referenced by: cicref 17073 cicsym 17076 cictr 17077 |
Copyright terms: Public domain | W3C validator |