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Theorem gicen 19198

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 19190	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4302	. . 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 19183	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6845	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8905	. . . . 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 217	. 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝐵 ≈ 𝐶)
11	1, 10	sylbi 217	1 ⊢ (𝑅 ≃_𝑔 𝑆 → 𝐵 ≈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2929 ∅c0 4282 class class class wbr 5095 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7355 ≈ cen 8876 Basecbs 17127 GrpIso cgim 19177 ≃_𝑔 cgic 19178

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7358 df-oprab 7359 df-mpo 7360 df-1st 7930 df-2nd 7931 df-1o 8394 df-map 8761 df-en 8880 df-ghm 19133 df-gim 19179 df-gic 19180

This theorem is referenced by: cyggic 21518 sconnpi1 35355

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