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Theorem lmimfn 19238
Description: Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Assertion
Ref Expression
lmimfn LMIso Fn (LMod × LMod)

Proof of Theorem lmimfn
Dummy variables 𝑠 𝑡 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmim 19235 . 2 LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2 ovex 6822 . . 3 (𝑠 LMHom 𝑡) ∈ V
32rabex 4943 . 2 {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
41, 3fnmpt2i 7388 1 LMIso Fn (LMod × LMod)
Colors of variables: wff setvar class
Syntax hints:  {crab 3064   × cxp 5247   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  Basecbs 16063  LModclmod 19072   LMHom clmhm 19231   LMIso clmim 19232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-lmim 19235
This theorem is referenced by:  brlmic  19280
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