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Theorem gicen 19543

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 19535	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4303	. . 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 19527	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵⟶1-1onto𝐶)
6	3	fvexi 6930	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 9037	. . . . 5 ⊢ (𝑓:𝐵⟶1-1onto𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1965	. . 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ≠ wne 2959 ∅c0 4281 class class class wbr 5108 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ≈ cen 9008 Basecbs 17431 GrpIso cgim 19521 ≃_𝑔 cgic 19522

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-1o 8514 df-map 8887 df-en 9012 df-ghm 19478 df-gim 19523 df-gic 19524

This theorem is used by: cyggic 21933 sconnpi1 36212

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