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Theorem opidon2OLD 35013
Description: Obsolete version of mndpfo 17922 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 2818 . . 3 dom dom 𝐺 = dom dom 𝐺
21opidonOLD 35011 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3 opidon2OLD.1 . . . 4 𝑋 = ran 𝐺
4 forn 6586 . . . 4 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
53, 4syl5req 2866 . . 3 (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6 xpeq12 5573 . . . . . . 7 ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
76anidms 567 . . . . . 6 (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8 foeq2 6580 . . . . . 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 6581 . . . . 5 (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
119, 10bitrd 280 . . . 4 (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺𝐺:(𝑋 × 𝑋)–onto𝑋))
1211biimpd 230 . . 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 207   = wceq 1528  wcel 2105  cin 3932   × cxp 5546  dom cdm 5548  ran crn 5549  ontowfo 6346   ExId cexid 35003  Magmacmagm 35007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-exid 35004  df-mgmOLD 35008
This theorem is referenced by:  exidreslem  35036
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