Theorem idrval 38192

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 30598	. 2 ⊢ (𝐺 ∈ 𝐴 → (GId‘𝐺) = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
4	1, 3	eqtrid 2784	1 ⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ran crn 5625  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  GIdcgi 30576

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7317  df-ov 7363  df-gid 30580

This theorem is referenced by:  iorlid  38193  cmpidelt  38194