Theorem lmimfn 20625

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.

Step	Hyp	Ref	Expression
1		df-lmim 20622	. 2 ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2		ovex 7437	. . 3 ⊢ (𝑠 LMHom 𝑡) ∈ V
3	2	rabex 5331	. 2 ⊢ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
4	1, 3	fnmpoi 8051	1 ⊢ LMIso Fn (LMod × LMod)

Syntax hints:  {crab 3433   × cxp 5673   Fn wfn 6535  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7404  Basecbs 17140  LModclmod 20459   LMHom clmhm 20618   LMIso clmim 20619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-lmim 20622

This theorem is referenced by:  brlmic  20667