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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 19224 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4347 | . . 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 19217 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶) |
6 | 3 | fvexi 6911 | . . . . . 6 ⊢ 𝐵 ∈ V |
7 | 6 | f1oen 8994 | . . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶) |
9 | 8 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶) |
10 | 2, 9 | sylbi 216 | . 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶) |
11 | 1, 10 | sylbi 216 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 class class class wbr 5148 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 ≈ cen 8961 Basecbs 17180 GrpIso cgim 19211 ≃𝑔 cgic 19212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-1o 8487 df-en 8965 df-ghm 19168 df-gim 19213 df-gic 19214 |
This theorem is referenced by: cyggic 21506 sconnpi1 34849 |
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