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Mirrors > Home > MPE Home > Th. List > lmimf1o | Structured version Visualization version GIF version |
Description: An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
islmim.b | ⊢ 𝐵 = (Base‘𝑅) |
islmim.c | ⊢ 𝐶 = (Base‘𝑆) |
Ref | Expression |
---|---|
lmimf1o | ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmim.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | islmim.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
3 | 1, 2 | islmim 19457 | . 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |
4 | 3 | simprbi 492 | 1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 LMHom clmhm 19414 LMIso clmim 19415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-lmhm 19417 df-lmim 19418 |
This theorem is referenced by: lmimgim 19460 lmimlbs 20579 lnmlmic 38617 |
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