Theorem isexid 35691

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.)

Hypothesis

Ref	Expression
isexid.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
isexid	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))

Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem isexid

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5757	. . . . 5 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2	1	dmeqd 5759	. . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3		isexid.1	. . . 4 ⊢ 𝑋 = dom dom 𝐺
4	2, 3	eqtr4di 2789	. . 3 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋)
5		oveq 7197	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
6	5	eqeq1d 2738	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦))
7		oveq 7197	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
8	7	eqeq1d 2738	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦))
9	6, 8	anbi12d 634	. . . 4 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
10	4, 9	raleqbidv 3303	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
11	4, 10	rexeqbidv 3304	. 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 35689	. 2 ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
13	11, 12	elab2g 3578	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  dom cdm 5536  (class class class)co 7191   ExId cexid 35688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-dm 5546  df-iota 6316  df-fv 6366  df-ov 7194  df-exid 35689

This theorem is referenced by:  opidonOLD  35696  isexid2  35699  ismndo  35716  exidres  35722