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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iorlid | Structured version Visualization version GIF version | ||
| Description: A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| iorlid.1 | ⊢ 𝑋 = ran 𝐺 | 
| iorlid.2 | ⊢ 𝑈 = (GId‘𝐺) | 
| Ref | Expression | 
|---|---|
| iorlid | ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iorlid.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
| 2 | iorlid.2 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
| 3 | 1, 2 | idrval 37865 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) | 
| 4 | 1 | exidu1 37864 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | 
| 5 | riotacl 7406 | . . 3 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ 𝑋) | 
| 7 | 3, 6 | eqeltrd 2840 | 1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃!wreu 3377 ∩ cin 3949 ran crn 5685 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 GIdcgi 30510 ExId cexid 37852 Magmacmagm 37856 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-riota 7389 df-ov 7435 df-gid 30514 df-exid 37853 df-mgmOLD 37857 | 
| This theorem is referenced by: cmpidelt 37867 rngo1cl 37947 | 
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