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Mirrors > Home > MPE Home > Th. List > gimf1o | Structured version Visualization version GIF version |
Description: An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
isgim.b | ⊢ 𝐵 = (Base‘𝑅) |
isgim.c | ⊢ 𝐶 = (Base‘𝑆) |
Ref | Expression |
---|---|
gimf1o | ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgim.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isgim.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
3 | 1, 2 | isgim 18666 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |
4 | 3 | simprbi 500 | 1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 GrpHom cghm 18619 GrpIso cgim 18661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-ghm 18620 df-gim 18663 |
This theorem is referenced by: subggim 18670 gicen 18681 gicsubgen 18682 giccyg 19285 gim0to0 19762 abliso 31024 gicabl 40627 |
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