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Theorem ismgmOLD 33779
Description: Obsolete version of ismgm 17290 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 (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Proof of Theorem ismgmOLD
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6064 . . . . 5 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
21exbidv 1890 . . . 4 (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3 df-mgmOLD 33778 . . . 4 Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
42, 3elab2g 3385 . . 3 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5 f00 6125 . . . . . . . 8 (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6 dmeq 5356 . . . . . . . . . 10 (𝐺 = ∅ → dom 𝐺 = dom ∅)
7 dmeq 5356 . . . . . . . . . . 11 (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8 dm0 5371 . . . . . . . . . . . . 13 dom ∅ = ∅
98dmeqi 5357 . . . . . . . . . . . 12 dom dom ∅ = dom ∅
109, 8eqtri 2673 . . . . . . . . . . 11 dom dom ∅ = ∅
117, 10syl6req 2702 . . . . . . . . . 10 (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
126, 11syl 17 . . . . . . . . 9 (𝐺 = ∅ → ∅ = dom dom 𝐺)
1312adantr 480 . . . . . . . 8 ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
145, 13sylbi 207 . . . . . . 7 (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15 xpeq12 5168 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
1615anidms 678 . . . . . . . . 9 (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17 feq23 6067 . . . . . . . . 9 (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
1816, 17mpancom 704 . . . . . . . 8 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
19 eqeq1 2655 . . . . . . . 8 (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
2018, 19imbi12d 333 . . . . . . 7 (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
2114, 20mpbiri 248 . . . . . 6 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
22 fdm 6089 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23 dmeq 5356 . . . . . . . 8 (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24 df-ne 2824 . . . . . . . . . . . 12 (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25 dmxp 5376 . . . . . . . . . . . 12 (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
2624, 25sylbir 225 . . . . . . . . . . 11 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
2726eqeq1d 2653 . . . . . . . . . 10 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺𝑡 = dom dom 𝐺))
2827biimpcd 239 . . . . . . . . 9 (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
2928eqcoms 2659 . . . . . . . 8 (dom dom 𝐺 = dom (𝑡 × 𝑡) → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3022, 23, 293syl 18 . . . . . . 7 (𝐺:(𝑡 × 𝑡)⟶𝑡 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
3130com12 32 . . . . . 6 𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
3221, 31pm2.61i 176 . . . . 5 (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺)
3332pm4.71ri 666 . . . 4 (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
3433exbii 1814 . . 3 (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
354, 34syl6bb 276 . 2 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡)))
36 dmexg 7139 . . 3 (𝐺𝐴 → dom 𝐺 ∈ V)
37 dmexg 7139 . . 3 (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38 xpeq12 5168 . . . . . . 7 ((𝑡 = dom dom 𝐺𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
3938anidms 678 . . . . . 6 (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40 feq23 6067 . . . . . 6 (((𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑡 = dom dom 𝐺) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
4139, 40mpancom 704 . . . . 5 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
42 ismgmOLD.1 . . . . . . . 8 𝑋 = dom dom 𝐺
4342eqcomi 2660 . . . . . . 7 dom dom 𝐺 = 𝑋
4443, 43xpeq12i 5171 . . . . . 6 (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
4544, 43feq23i 6077 . . . . 5 (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋)
4641, 45syl6bb 276 . . . 4 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
4746ceqsexgv 3366 . . 3 (dom dom 𝐺 ∈ V → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4836, 37, 473syl 18 . 2 (𝐺𝐴 → (∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
4935, 48bitrd 268 1 (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  c0 3948   × cxp 5141  dom cdm 5143  wf 5922  Magmacmagm 33777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-mgmOLD 33778
This theorem is referenced by:  clmgmOLD  33780  opidonOLD  33781  issmgrpOLD  33792
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