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Mirrors > Home > MPE Home > Th. List > gicer | Structured version Visualization version GIF version |
Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.) |
Ref | Expression |
---|---|
gicer | ⊢ ≃𝑔 Er Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gic 18402 | . . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o)) | |
2 | cnvimass 5951 | . . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso | |
3 | gimfn 18403 | . . . . . 6 ⊢ GrpIso Fn (Grp × Grp) | |
4 | fndm 6457 | . . . . . 6 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ dom GrpIso = (Grp × Grp) |
6 | 2, 5 | sseqtri 4005 | . . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp) |
7 | 1, 6 | eqsstri 4003 | . . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp) |
8 | relxp 5575 | . . 3 ⊢ Rel (Grp × Grp) | |
9 | relss 5658 | . . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 )) | |
10 | 7, 8, 9 | mp2 9 | . 2 ⊢ Rel ≃𝑔 |
11 | gicsym 18416 | . 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥) | |
12 | gictr 18417 | . 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧) | |
13 | gicref 18413 | . . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥) | |
14 | giclcl 18414 | . . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp) | |
15 | 13, 14 | impbii 211 | . 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥) |
16 | 10, 11, 12, 15 | iseri 8318 | 1 ⊢ ≃𝑔 Er Grp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 class class class wbr 5068 × cxp 5555 ◡ccnv 5556 dom cdm 5557 “ cima 5560 Rel wrel 5562 Fn wfn 6352 1oc1o 8097 Er wer 8288 Grpcgrp 18105 GrpIso cgim 18399 ≃𝑔 cgic 18400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-map 8410 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-ghm 18358 df-gim 18401 df-gic 18402 |
This theorem is referenced by: (None) |
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