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Mirrors > Home > MPE Home > Th. List > brgic | Structured version Visualization version GIF version |
Description: The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
brgic | ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gic 18338 | . 2 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o)) | |
2 | gimfn 18339 | . 2 ⊢ GrpIso Fn (Grp × Grp) | |
3 | 1, 2 | brwitnlem 8121 | 1 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 × cxp 5546 (class class class)co 7145 Grpcgrp 18041 GrpIso cgim 18335 ≃𝑔 cgic 18336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-1o 8091 df-gim 18337 df-gic 18338 |
This theorem is referenced by: brgici 18348 giclcl 18350 gicrcl 18351 gicsym 18352 gictr 18353 gicen 18355 gicsubgen 18356 giccyg 18949 gicabl 39577 |
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