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Theorem lmimf1o 20908
Description: An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b 𝐵 = (Base‘𝑅)
islmim.c 𝐶 = (Base‘𝑆)
Assertion
Ref Expression
lmimf1o (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)

Proof of Theorem lmimf1o
StepHypRef Expression
1 islmim.b . . 3 𝐵 = (Base‘𝑅)
2 islmim.c . . 3 𝐶 = (Base‘𝑆)
31, 2islmim 20907 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
43simprbi 496 1 (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  1-1-ontowf1o 6535  cfv 6536  (class class class)co 7404  Basecbs 17150   LMHom clmhm 20864   LMIso clmim 20865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-lmhm 20867  df-lmim 20868
This theorem is referenced by:  lmimgim  20910  lmimlbs  21726  lmimdim  33205  lnmlmic  42390
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