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Definition df-lmim 19245
Description: An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
df-lmim LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
Distinct variable group:   𝑡,𝑠,𝑔

Detailed syntax breakdown of Definition df-lmim
StepHypRef Expression
1 clmim 19242 . 2 class LMIso
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 clmod 19085 . . 3 class LMod
52cv 1631 . . . . . 6 class 𝑠
6 cbs 16079 . . . . . 6 class Base
75, 6cfv 6049 . . . . 5 class (Base‘𝑠)
83cv 1631 . . . . . 6 class 𝑡
98, 6cfv 6049 . . . . 5 class (Base‘𝑡)
10 vg . . . . . 6 setvar 𝑔
1110cv 1631 . . . . 5 class 𝑔
127, 9, 11wf1o 6048 . . . 4 wff 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)
13 clmhm 19241 . . . . 5 class LMHom
145, 8, 13co 6814 . . . 4 class (𝑠 LMHom 𝑡)
1512, 10, 14crab 3054 . . 3 class {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}
162, 3, 4, 4, 15cmpt2 6816 . 2 class (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
171, 16wceq 1632 1 wff LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
Colors of variables: wff setvar class
This definition is referenced by:  lmimfn  19248  islmim  19284
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