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Theorem ismgmOLD 34135
Description: Obsolete version of ismgm 17557 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 6238 . . . . 5 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
21exbidv 2017 . . . 4 (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3 df-mgmOLD 34134 . . . 4 Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
42, 3elab2g 3546 . . 3 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5 f00 6303 . . . . . . . 8 (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6 dmeq 5528 . . . . . . . . . 10 (𝐺 = ∅ → dom 𝐺 = dom ∅)
7 dmeq 5528 . . . . . . . . . . 11 (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8 dm0 5543 . . . . . . . . . . . . 13 dom ∅ = ∅
98dmeqi 5529 . . . . . . . . . . . 12 dom dom ∅ = dom ∅
109, 8eqtri 2822 . . . . . . . . . . 11 dom dom ∅ = ∅
117, 10syl6req 2851 . . . . . . . . . 10 (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
126, 11syl 17 . . . . . . . . 9 (𝐺 = ∅ → ∅ = dom dom 𝐺)
1312adantr 473 . . . . . . . 8 ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
145, 13sylbi 209 . . . . . . 7 (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15 xpeq12 5338 . . . . . . . . . 10 ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
1615anidms 563 . . . . . . . . 9 (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17 feq23 6241 . . . . . . . . 9 (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
1816, 17mpancom 680 . . . . . . . 8 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(∅ × ∅)⟶∅))
19 eqeq1 2804 . . . . . . . 8 (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
2018, 19imbi12d 336 . . . . . . 7 (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
2114, 20mpbiri 250 . . . . . 6 (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺))
22 fdm 6265 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23 dmeq 5528 . . . . . . . 8 (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24 df-ne 2973 . . . . . . . . . . . 12 (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25 dmxp 5548 . . . . . . . . . . . 12 (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
2624, 25sylbir 227 . . . . . . . . . . 11 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
2726eqeq1d 2802 . . . . . . . . . 10 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺𝑡 = dom dom 𝐺))
2827biimpcd 241 . . . . . . . . 9 (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
2928eqcoms 2808 . . . . . . . 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 177 . . . . 5 (𝐺:(𝑡 × 𝑡)⟶𝑡𝑡 = dom dom 𝐺)
3332pm4.71ri 557 . . . 4 (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
3433exbii 1944 . . 3 (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡))
354, 34syl6bb 279 . 2 (𝐺𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺𝐺:(𝑡 × 𝑡)⟶𝑡)))
36 dmexg 7332 . . 3 (𝐺𝐴 → dom 𝐺 ∈ V)
37 dmexg 7332 . . 3 (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38 xpeq12 5338 . . . . . . 7 ((𝑡 = dom dom 𝐺𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
3938anidms 563 . . . . . 6 (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40 feq23 6241 . . . . . 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 680 . . . . 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 2809 . . . . . . 7 dom dom 𝐺 = 𝑋
4443, 43xpeq12i 5341 . . . . . 6 (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
4544, 43feq23i 6251 . . . . 5 (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋)
4641, 45syl6bb 279 . . . 4 (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
4746ceqsexgv 3525 . . 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 271 1 (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wne 2972  Vcvv 3386  c0 4116   × cxp 5311  dom cdm 5313  wf 6098  Magmacmagm 34133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-fun 6104  df-fn 6105  df-f 6106  df-mgmOLD 34134
This theorem is referenced by:  clmgmOLD  34136  opidonOLD  34137  issmgrpOLD  34148
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