gicref - Metamath Proof Explorer

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Theorem gicref 19466

Description: Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Assertion**
Ref	Expression
gicref	⊢ (𝑅 ∈ Grp → 𝑅 ≃_𝑔 𝑅)

**Proof of Theorem gicref**
Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2	1	idghm 19425	. . 3 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
3		cnvresid 6611	. . . 4 ⊢ ^◡( I ↾ (Base‘𝑅)) = ( I ↾ (Base‘𝑅))
4	3, 2	eqeltrid 2865	. . 3 ⊢ (𝑅 ∈ Grp → ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅))
5		isgim2 19459	. . 3 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) ↔ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅) ∧ ^◡( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpHom 𝑅)))
6	2, 4, 5	sylanbrc 595	. 2 ⊢ (𝑅 ∈ Grp → ( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅))
7		brgici 19465	. 2 ⊢ (( I ↾ (Base‘𝑅)) ∈ (𝑅 GrpIso 𝑅) → 𝑅 ≃_𝑔 𝑅)
8	6, 7	syl 18	1 ⊢ (𝑅 ∈ Grp → 𝑅 ≃_𝑔 𝑅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 class class class wbr 5103 I cid 5545 ^◡ccnv 5650 ↾ cres 5653 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 Grpcgrp 19124 GrpHom cghm 19407 GrpIso cgim 19451 ≃_𝑔 cgic 19452

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 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-1o 8460 df-map 8833 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-grp 19127 df-ghm 19408 df-gim 19453 df-gic 19454

This theorem is used by: gicer 19471

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