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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 18409 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4310 | . . 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 18403 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶) |
6 | 3 | fvexi 6684 | . . . . . 6 ⊢ 𝐵 ∈ V |
7 | 6 | f1oen 8530 | . . . . 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 219 | . 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶) |
11 | 1, 10 | sylbi 219 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ≈ cen 8506 Basecbs 16483 GrpIso cgim 18397 ≃𝑔 cgic 18398 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-1o 8102 df-en 8510 df-ghm 18356 df-gim 18399 df-gic 18400 |
This theorem is referenced by: cyggic 20719 sconnpi1 32486 |
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