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Theorem gicen 19405

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 19397	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4300	. . 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 19390	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓:𝐵–1-1-onto→𝐶)
6	3	fvexi 6892	. . . . . 6 ⊢ 𝐵 ∈ V
7	6	f1oen 8978	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐶 → 𝐵 ≈ 𝐶)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝐵 ≈ 𝐶)
9	8	exlimiv 1963	. . 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 2145 ≠ wne 2955 ∅c0 4279 class class class wbr 5103 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7413 ≈ cen 8949 Basecbs 17301 GrpIso cgim 19384 ≃_𝑔 cgic 19385

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-1o 8455 df-map 8828 df-en 8953 df-ghm 19341 df-gim 19386 df-gic 19387

This theorem is used by: cyggic 21785 sconnpi1 35818

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