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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 19293 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |
4 | 3 | simprbi 496 | 1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 GrpHom cghm 19243 GrpIso cgim 19288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ghm 19244 df-gim 19290 |
This theorem is referenced by: subggim 19297 gim0to0 19300 gicen 19309 gicsubgen 19310 giccyg 19933 abliso 33024 lmhmqusker 33425 rhmqusker 33434 aks6d1c6lem5 42159 gicabl 43088 |
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