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Mirrors > Home > MPE Home > Th. List > gidval | Structured version Visualization version GIF version |
Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
gidval.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
gidval | ⊢ (𝐺 ∈ 𝑉 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
2 | rneq 5272 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
3 | gidval.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 2, 3 | syl6eqr 2662 | . . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
5 | oveq 6555 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥)) | |
6 | 5 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥)) |
7 | oveq 6555 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢)) | |
8 | 7 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥)) |
9 | 6, 8 | anbi12d 743 | . . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
10 | 4, 9 | raleqbidv 3129 | . . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
11 | 4, 10 | riotaeqbidv 6514 | . . 3 ⊢ (𝑔 = 𝐺 → (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
12 | df-gid 26732 | . . 3 ⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥))) | |
13 | riotaex 6515 | . . 3 ⊢ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V | |
14 | 11, 12, 13 | fvmpt 6191 | . 2 ⊢ (𝐺 ∈ V → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
15 | 1, 14 | syl 17 | 1 ⊢ (𝐺 ∈ 𝑉 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ran crn 5039 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 GIdcgi 26728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-gid 26732 |
This theorem is referenced by: grpoidval 26751 idrval 32826 exidresid 32848 |
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