Theorem ismgmOLD 37879

Description: Obsolete version of ismgm 18624 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ismgmOLD.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
ismgmOLD	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Proof of Theorem ismgmOLD

Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 6691	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑡 × 𝑡)⟶𝑡))
2	1	exbidv 1921	. . . 4 ⊢ (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3		df-mgmOLD 37878	. . . 4 ⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
4	2, 3	elab2g 3664	. . 3 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5		f00 6765	. . . . . . . 8 ⊢ (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6		dmeq 5888	. . . . . . . . . 10 ⊢ (𝐺 = ∅ → dom 𝐺 = dom ∅)
7		dmeq 5888	. . . . . . . . . . 11 ⊢ (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8		dm0 5905	. . . . . . . . . . . . 13 ⊢ dom ∅ = ∅
9	8	dmeqi 5889	. . . . . . . . . . . 12 ⊢ dom dom ∅ = dom ∅
10	9, 8	eqtri 2759	. . . . . . . . . . 11 ⊢ dom dom ∅ = ∅
11	7, 10	eqtr2di 2788	. . . . . . . . . 10 ⊢ (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (𝐺 = ∅ → ∅ = dom dom 𝐺)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
14	5, 13	sylbi 217	. . . . . . 7 ⊢ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15		xpeq12 5684	. . . . . . . . . 10 ⊢ ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
16	15	anidms 566	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17		feq23 6694	. . . . . . . . 9 ⊢ (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(∅ × ∅)⟶∅))
18	16, 17	mpancom 688	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(∅ × ∅)⟶∅))
19		eqeq1 2740	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
20	18, 19	imbi12d 344	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
21	14, 20	mpbiri 258	. . . . . 6 ⊢ (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺))
22		fdm 6720	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23		dmeq 5888	. . . . . . . 8 ⊢ (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24		df-ne 2934	. . . . . . . . . . . 12 ⊢ (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25		dmxp 5913	. . . . . . . . . . . 12 ⊢ (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
26	24, 25	sylbir 235	. . . . . . . . . . 11 ⊢ (¬ 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
27	26	eqeq1d 2738	. . . . . . . . . 10 ⊢ (¬ 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺 ↔ 𝑡 = dom dom 𝐺))
28	27	biimpcd 249	. . . . . . . . 9 ⊢ (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
29	28	eqcoms 2744	. . . . . . . 8 ⊢ (dom dom 𝐺 = dom (𝑡 × 𝑡) → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
30	22, 23, 29	3syl 18	. . . . . . 7 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
31	30	com12 32	. . . . . 6 ⊢ (¬ 𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺))
32	21, 31	pm2.61i 182	. . . . 5 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺)
33	32	pm4.71ri 560	. . . 4 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡))
34	33	exbii 1848	. . 3 ⊢ (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡))
35	4, 34	bitrdi 287	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡)))
36		dmexg 7902	. . 3 ⊢ (𝐺 ∈ 𝐴 → dom 𝐺 ∈ V)
37		dmexg 7902	. . 3 ⊢ (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38		xpeq12 5684	. . . . . . 7 ⊢ ((𝑡 = dom dom 𝐺 ∧ 𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
39	38	anidms 566	. . . . . 6 ⊢ (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40		feq23 6694	. . . . . 6 ⊢ (((𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑡 = dom dom 𝐺) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
41	39, 40	mpancom 688	. . . . 5 ⊢ (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
42		ismgmOLD.1	. . . . . . . 8 ⊢ 𝑋 = dom dom 𝐺
43	42	eqcomi 2745	. . . . . . 7 ⊢ dom dom 𝐺 = 𝑋
44	43, 43	xpeq12i 5687	. . . . . 6 ⊢ (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
45	44, 43	feq23i 6705	. . . . 5 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋)
46	41, 45	bitrdi 287	. . . 4 ⊢ (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
47	46	ceqsexgv 3638	. . 3 ⊢ (dom dom 𝐺 ∈ V → (∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
48	36, 37, 47	3syl 18	. 2 ⊢ (𝐺 ∈ 𝐴 → (∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
49	35, 48	bitrd 279	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464  ∅c0 4313   × cxp 5657  dom cdm 5659  ⟶wf 6532  Magmacmagm 37877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6538  df-fn 6539  df-f 6540  df-mgmOLD 37878

This theorem is referenced by:  clmgmOLD  37880  opidonOLD  37881  issmgrpOLD  37892