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Mirrors > Home > MPE Home > Th. List > isgim2 | Structured version Visualization version GIF version |
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 23232. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
isgim2 | ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2733 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | 1, 2 | isgim 19121 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) |
4 | 1, 2 | ghmf1o 19107 | . . 3 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) |
5 | 4 | pm5.32i 576 | . 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) |
6 | 3, 5 | bitri 275 | 1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ◡ccnv 5671 –1-1-onto→wf1o 6534 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 GrpHom cghm 19074 GrpIso cgim 19116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-ghm 19075 df-gim 19118 |
This theorem is referenced by: gimcnv 19126 gimco 19127 gicref 19130 pi1xfrgim 24543 |
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