Theorem isexid 38050

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 5853	. . . . 5 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2	1	dmeqd 5855	. . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3		isexid.1	. . . 4 ⊢ 𝑋 = dom dom 𝐺
4	2, 3	eqtr4di 2790	. . 3 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = 𝑋)
5		oveq 7366	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
6	5	eqeq1d 2739	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = 𝑦 ↔ (𝑥𝐺𝑦) = 𝑦))
7		oveq 7366	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
8	7	eqeq1d 2739	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑦 ↔ (𝑦𝐺𝑥) = 𝑦))
9	6, 8	anbi12d 633	. . . 4 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
10	4, 9	raleqbidv 3317	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
11	4, 10	rexeqbidv 3318	. 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 38048	. 2 ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
13	11, 12	elab2g 3636	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  dom cdm 5625  (class class class)co 7360   ExId cexid 38047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-dm 5635  df-iota 6449  df-fv 6501  df-ov 7363  df-exid 38048

This theorem is referenced by:  opidonOLD  38055  isexid2  38058  ismndo  38075  exidres  38081