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Theorem opidon2OLD 32823
Description: Obsolete version of mndpfo 17137 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 2610 . . 3 dom dom 𝐺 = dom dom 𝐺
21opidonOLD 32821 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3 opidon2OLD.1 . . . 4 𝑋 = ran 𝐺
4 forn 6031 . . . 4 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
53, 4syl5req 2657 . . 3 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6 xpeq12 5058 . . . . . . 7 ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
76anidms 675 . . . . . 6 (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8 foeq2 6025 . . . . . 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 6026 . . . . 5 (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
119, 10bitrd 267 . . . 4 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
1211biimpd 218 . . 3 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
135, 12mpcom 37 . 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 195   = wceq 1475  wcel 1977  cin 3539   × cxp 5036  dom cdm 5038  ran crn 5039  ontowfo 5802   ExId cexid 32813  Magmacmagm 32817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-exid 32814  df-mgmOLD 32818
This theorem is referenced by:  exidreslem  32846
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