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Mirrors > Home > MPE Home > Th. List > Mathboxes > isexid | Structured version Visualization version GIF version |
Description: The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isexid.1 | ⊢ 𝑋 = dom dom 𝐺 |
Ref | Expression |
---|---|
isexid | ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5618 | . . . . 5 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
2 | 1 | dmeqd 5620 | . . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺) |
3 | isexid.1 | . . . 4 ⊢ 𝑋 = dom dom 𝐺 | |
4 | 2, 3 | syl6eqr 2825 | . . 3 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋) |
5 | oveq 6980 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦)) | |
6 | 5 | eqeq1d 2773 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦)) |
7 | oveq 6980 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
8 | 7 | eqeq1d 2773 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦)) |
9 | 6, 8 | anbi12d 622 | . . . 4 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
10 | 4, 9 | raleqbidv 3334 | . . 3 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
11 | 4, 10 | rexeqbidv 3335 | . 2 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
12 | df-exid 34602 | . 2 ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)} | |
13 | 11, 12 | elab2g 3577 | 1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ∃wrex 3082 dom cdm 5403 (class class class)co 6974 ExId cexid 34601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-dm 5413 df-iota 6149 df-fv 6193 df-ov 6977 df-exid 34602 |
This theorem is referenced by: opidonOLD 34609 isexid2 34612 ismndo 34629 exidres 34635 |
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