Theorem idrval 37881

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	⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((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

Step	Hyp	Ref	Expression
1		idrval.2	. 2 ⊢ 𝑈 = (GId‘𝐺)
2		idrval.1	. . 3 ⊢ 𝑋 = ran 𝐺
3	2	gidval 30493	. 2 ⊢ (𝐺 ∈ 𝐴 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4	1, 3	eqtrid 2782	1 ⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ran crn 5655  ‘cfv 6531  ℩crio 7361  (class class class)co 7405  GIdcgi 30471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-riota 7362  df-ov 7408  df-gid 30475

This theorem is referenced by:  iorlid  37882  cmpidelt  37883