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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 3212 | . 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
2 | rneq 5351 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
3 | gidval.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 2, 3 | syl6eqr 2674 | . . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
5 | oveq 6656 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥)) | |
6 | 5 | eqeq1d 2624 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥)) |
7 | oveq 6656 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑢) = (𝑥𝐺𝑢)) | |
8 | 7 | eqeq1d 2624 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑢) = 𝑥 ↔ (𝑥𝐺𝑢) = 𝑥)) |
9 | 6, 8 | anbi12d 747 | . . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
10 | 4, 9 | raleqbidv 3152 | . . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
11 | 4, 10 | riotaeqbidv 6614 | . . 3 ⊢ (𝑔 = 𝐺 → (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) |
12 | df-gid 27348 | . . 3 ⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥))) | |
13 | riotaex 6615 | . . 3 ⊢ (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ∈ V | |
14 | 11, 12, 13 | fvmpt 6282 | . 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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ran crn 5115 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 GIdcgi 27344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-gid 27348 |
This theorem is referenced by: grpoidval 27367 idrval 33656 exidresid 33678 |
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