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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 37864 | . 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) | 
| 4 | 1 | exidu1 37863 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | 
| 5 | riotacl 7405 | . . 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 2841 | 1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 ∩ cin 3950 ran crn 5686 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 GIdcgi 30509 ExId cexid 37851 Magmacmagm 37855 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-riota 7388 df-ov 7434 df-gid 30513 df-exid 37852 df-mgmOLD 37856 | 
| This theorem is referenced by: cmpidelt 37866 rngo1cl 37946 | 
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