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Mirrors > Home > MPE Home > Th. List > gicen | Structured version Visualization version GIF version |
Description: Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
gicen.b | ⊢ 𝐵 = (Base‘𝑅) |
gicen.c | ⊢ 𝐶 = (Base‘𝑆) |
Ref | Expression |
---|---|
gicen | ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 18401 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4260 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
3 | gicen.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | gicen.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑆) | |
5 | 3, 4 | gimf1o 18395 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶) |
6 | 3 | fvexi 6659 | . . . . . 6 ⊢ 𝐵 ∈ V |
7 | 6 | f1oen 8513 | . . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶) |
9 | 8 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶) |
10 | 2, 9 | sylbi 220 | . 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶) |
11 | 1, 10 | sylbi 220 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 class class class wbr 5030 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ≈ cen 8489 Basecbs 16475 GrpIso cgim 18389 ≃𝑔 cgic 18390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-1o 8085 df-en 8493 df-ghm 18348 df-gim 18391 df-gic 18392 |
This theorem is referenced by: cyggic 20264 sconnpi1 32599 |
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