Theorem idrval 39057

Description: Obsolete theorem, use idvalriota 18891 instead. The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification 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 31171	. 2 ⊢ (𝐺 ∈ 𝐴 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4	1, 3	eqtrid 2810	1 ⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ran crn 5659  ‘cfv 6566  ℩crio 7407  (class class class)co 7451  GIdcgi 31149

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-riota 7408  df-ov 7454  df-gid 31153

This theorem is used by:  iorlid  39058  cmpidelt  39059