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Metamath Proof Explorer
Description: The value of the identity element. (Contributed by FL, 12-Dec-2009)
(Revised by Mario Carneiro, 22-Dec-2013)
(New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
idrval.1 |
⊢ 𝑋 = ran 𝐺 |
|
|
idrval.2 |
⊢ 𝑈 = ( GId ‘ 𝐺 ) |
|
Assertion |
idrval |
⊢ ( 𝐺 ∈ 𝐴 → 𝑈 = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
idrval.1 |
⊢ 𝑋 = ran 𝐺 |
| 2 |
|
idrval.2 |
⊢ 𝑈 = ( GId ‘ 𝐺 ) |
| 3 |
1
|
gidval |
⊢ ( 𝐺 ∈ 𝐴 → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) ) |
| 4 |
2 3
|
eqtrid |
⊢ ( 𝐺 ∈ 𝐴 → 𝑈 = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) ) |