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Theorem ismgmOLD 33881
Description: Obsolete version of ismgm 17365 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 6139 . . . . 5 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
21exbidv 1963 . . . 4 (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3 df-mgmOLD 33880 . . . 4 Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
42, 3elab2g 3458 . . 3 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5 f00 6200 . . . . . . . 8 (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6 dmeq 5431 . . . . . . . . . 10 (𝐺 = ∅ → dom 𝐺 = dom ∅)
7 dmeq 5431 . . . . . . . . . . 11 (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8 dm0 5446 . . . . . . . . . . . . 13 dom ∅ = ∅
98dmeqi 5432 . . . . . . . . . . . 12 dom dom ∅ = dom ∅
109, 8eqtri 2746 . . . . . . . . . . 11 dom dom ∅ = ∅
117, 10syl6req 2775 . . . . . . . . . 10 (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
126, 11syl 17 . . . . . . . . 9 (𝐺 = ∅ → ∅ = dom dom 𝐺)
1312adantr 472 . . . . . . . 8 ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
145, 13sylbi 207 . . . . . . 7 (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15 xpeq12 5243 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
1615anidms 680 . . . . . . . . 9 (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17 feq23 6142 . . . . . . . . 9 (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
1816, 17mpancom 706 . . . . . . . 8 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
19 eqeq1 2728 . . . . . . . 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 6164 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23 dmeq 5431 . . . . . . . 8 (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24 df-ne 2897 . . . . . . . . . . . 12 (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25 dmxp 5451 . . . . . . . . . . . 12 (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
2624, 25sylbir 225 . . . . . . . . . . 11 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
2726eqeq1d 2726 . . . . . . . . . 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 2732 . . . . . . . 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 668 . . . 4 (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
3433exbii 1887 . . 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 7214 . . 3 (𝐺𝐴 → dom 𝐺 ∈ V)
37 dmexg 7214 . . 3 (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38 xpeq12 5243 . . . . . . 7 ((𝑡 = dom dom 𝐺𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
3938anidms 680 . . . . . 6 (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40 feq23 6142 . . . . . 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 706 . . . . 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 2733 . . . . . . 7 dom dom 𝐺 = 𝑋
4443, 43xpeq12i 5246 . . . . . 6 (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
4544, 43feq23i 6152 . . . . 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 3439 . . 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 1596  wex 1817  wcel 2103  wne 2896  Vcvv 3304  c0 4023   × cxp 5216  dom cdm 5218  wf 5997  Magmacmagm 33879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-fun 6003  df-fn 6004  df-f 6005  df-mgmOLD 33880
This theorem is referenced by:  clmgmOLD  33882  opidonOLD  33883  issmgrpOLD  33894
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