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Theorem opidon2OLD 33783
Description: Obsolete version of mndpfo 17361 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
opidon2OLD.1 𝑋 = ran 𝐺
Assertion
Ref Expression
opidon2OLD (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)

Proof of Theorem opidon2OLD
StepHypRef Expression
1 eqid 2651 . . 3 dom dom 𝐺 = dom dom 𝐺
21opidonOLD 33781 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3 opidon2OLD.1 . . . 4 𝑋 = ran 𝐺
4 forn 6156 . . . 4 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
53, 4syl5req 2698 . . 3 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6 xpeq12 5168 . . . . . . 7 ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
76anidms 678 . . . . . 6 (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8 foeq2 6150 . . . . . 6 ((dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋) → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺))
97, 8syl 17 . . . . 5 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺))
10 foeq3 6151 . . . . 5 (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
119, 10bitrd 268 . . . 4 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
1211biimpd 219 . . 3 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
135, 12mpcom 38 . 2 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋)
142, 13syl 17 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  cin 3606   × cxp 5141  dom cdm 5143  ran crn 5144  ontowfo 5924   ExId cexid 33773  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-sbc 3469  df-csb 3567  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-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-exid 33774  df-mgmOLD 33778
This theorem is referenced by:  exidreslem  33806
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