Theorem lmimfn 21128

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 21125	. 2 ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2		ovex 7445	. . 3 ⊢ (𝑠 LMHom 𝑡) ∈ V
3	2	rabex 5311	. 2 ⊢ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
4	1, 3	fnmpoi 8068	1 ⊢ LMIso Fn (LMod × LMod)

Syntax hints:  {crab 3416   × cxp 5661   Fn wfn 6533  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  LModclmod 20962   LMHom clmhm 21121   LMIso clmim 21122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-lmim 21125

This theorem is referenced by:  brlmic  21170