Theorem lmimfn 20933

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 20930	. 2 ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2		ovex 7420	. . 3 ⊢ (𝑠 LMHom 𝑡) ∈ V
3	2	rabex 5294	. 2 ⊢ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
4	1, 3	fnmpoi 8049	1 ⊢ LMIso Fn (LMod × LMod)

Syntax hints:  {crab 3405   × cxp 5636   Fn wfn 6506  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  LModclmod 20766   LMHom clmhm 20926   LMIso clmim 20927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-lmim 20930

This theorem is referenced by:  brlmic  20975