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Theorem idrval 36720
Description: The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
idrval.1 𝑋 = ran 𝐺
idrval.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
idrval (𝐺𝐴𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑢)   𝑈(𝑥,𝑢)

Proof of Theorem idrval
StepHypRef Expression
1 idrval.2 . 2 𝑈 = (GId‘𝐺)
2 idrval.1 . . 3 𝑋 = ran 𝐺
32gidval 29760 . 2 (𝐺𝐴 → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
41, 3eqtrid 2784 1 (𝐺𝐴𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  ran crn 5677  cfv 6543  crio 7363  (class class class)co 7408  GIdcgi 29738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7364  df-ov 7411  df-gid 29742
This theorem is referenced by:  iorlid  36721  cmpidelt  36722
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