Theorem lmimf1o 19837

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:𝐵–1-1-onto→𝐶)

Proof of Theorem lmimf1o

Step	Hyp	Ref	Expression
1		islmim.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		islmim.c	. . 3 ⊢ 𝐶 = (Base‘𝑆)
3	1, 2	islmim 19836	. 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
4	3	simprbi 499	1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Basecbs 16485   LMHom clmhm 19793   LMIso clmim 19794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-lmhm 19796  df-lmim 19797

This theorem is referenced by:  lmimgim  19839  lmimlbs  20982  lnmlmic  39695