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Theorem gicen 19309

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 19301	. 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 19294	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6876	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8947	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1949	. . 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 1559 ∃wex 1798 ∈ wcel 2141 ≠ wne 2956 ∅c0 4283 class class class wbr 5097 –1-1-onto→wf1o 6515 ‘cfv 6516 (class class class)co 7391 ≈ cen 8918 Basecbs 17236 GrpIso cgim 19288 ≃_𝑔 cgic 19289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-1st 7965 df-2nd 7966 df-1o 8431 df-map 8804 df-en 8922 df-ghm 19245 df-gim 19290 df-gic 19291

This theorem is referenced by: cyggic 21612 sconnpi1 35550

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