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Theorem opidon2OLD 33935
Description: Obsolete version of mndpfo 17486 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 2748 . . 3 dom dom 𝐺 = dom dom 𝐺
21opidonOLD 33933 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3 opidon2OLD.1 . . . 4 𝑋 = ran 𝐺
4 forn 6267 . . . 4 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
53, 4syl5req 2795 . . 3 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6 xpeq12 5279 . . . . . . 7 ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
76anidms 680 . . . . . 6 (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8 foeq2 6261 . . . . . 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 6262 . . . . 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 1620  wcel 2127  cin 3702   × cxp 5252  dom cdm 5254  ran crn 5255  ontowfo 6035   ExId cexid 33925  Magmacmagm 33929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fo 6043  df-fv 6045  df-ov 6804  df-exid 33926  df-mgmOLD 33930
This theorem is referenced by:  exidreslem  33958
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