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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 20959 | . 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |
4 | 3 | simprbi 495 | 1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 LMHom clmhm 20916 LMIso clmim 20917 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-lmhm 20919 df-lmim 20920 |
This theorem is referenced by: lmimgim 20962 lmimlbs 21787 lmimdim 33432 lnmlmic 42654 |
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