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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 5917 | . . . . 5 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
2 | 1 | dmeqd 5919 | . . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺) |
3 | isexid.1 | . . . 4 ⊢ 𝑋 = dom dom 𝐺 | |
4 | 2, 3 | eqtr4di 2793 | . . 3 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋) |
5 | oveq 7437 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦)) | |
6 | 5 | eqeq1d 2737 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦)) |
7 | oveq 7437 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
8 | 7 | eqeq1d 2737 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦)) |
9 | 6, 8 | anbi12d 632 | . . . 4 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
10 | 4, 9 | raleqbidv 3344 | . . 3 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
11 | 4, 10 | rexeqbidv 3345 | . 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 37832 | . 2 ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)} | |
13 | 11, 12 | elab2g 3683 | 1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 dom cdm 5689 (class class class)co 7431 ExId cexid 37831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-exid 37832 |
This theorem is referenced by: opidonOLD 37839 isexid2 37842 ismndo 37859 exidres 37865 |
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