gicen - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gicen		Structured version Visualization version GIF version

Theorem gicen 18893

Description: Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)

**Hypotheses**
Ref	Expression
gicen.b	⊢ 𝐵 = (Base‘𝑅)
gicen.c	⊢ 𝐶 = (Base‘𝑆)

**Assertion**
Ref	Expression
gicen	⊢ (𝑅 ≃_𝑔 𝑆 → 𝐵 ≈ 𝐶)

Proof of Theorem gicen Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgic 18885	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4280	. . 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 18879	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6788	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8761	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1933	. . 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 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 class class class wbr 5074 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ≈ cen 8730 Basecbs 16912 GrpIso cgim 18873 ≃_𝑔 cgic 18874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-1o 8297 df-en 8734 df-ghm 18832 df-gim 18875 df-gic 18876

This theorem is referenced by: cyggic 20780 sconnpi1 33201

W3C validator