Metamath Proof Explorer


Theorem isof1oidb

Description: A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021)

Ref Expression
Assertion isof1oidb ( 𝐻 : 𝐴1-1-onto𝐵𝐻 Isom I , I ( 𝐴 , 𝐵 ) )

Proof

Step Hyp Ref Expression
1 f1of1 ( 𝐻 : 𝐴1-1-onto𝐵𝐻 : 𝐴1-1𝐵 )
2 f1fveq ( ( 𝐻 : 𝐴1-1𝐵 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝐻𝑥 ) = ( 𝐻𝑦 ) ↔ 𝑥 = 𝑦 ) )
3 1 2 sylan ( ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝐻𝑥 ) = ( 𝐻𝑦 ) ↔ 𝑥 = 𝑦 ) )
4 fvex ( 𝐻𝑦 ) ∈ V
5 4 ideq ( ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ↔ ( 𝐻𝑥 ) = ( 𝐻𝑦 ) )
6 5 a1i ( ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ↔ ( 𝐻𝑥 ) = ( 𝐻𝑦 ) ) )
7 ideqg ( 𝑦𝐴 → ( 𝑥 I 𝑦𝑥 = 𝑦 ) )
8 7 ad2antll ( ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥 I 𝑦𝑥 = 𝑦 ) )
9 3 6 8 3bitr4rd ( ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥 I 𝑦 ↔ ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ) )
10 9 ralrimivva ( 𝐻 : 𝐴1-1-onto𝐵 → ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ) )
11 10 pm4.71i ( 𝐻 : 𝐴1-1-onto𝐵 ↔ ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ) ) )
12 df-isom ( 𝐻 Isom I , I ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴1-1-onto𝐵 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻𝑥 ) I ( 𝐻𝑦 ) ) ) )
13 11 12 bitr4i ( 𝐻 : 𝐴1-1-onto𝐵𝐻 Isom I , I ( 𝐴 , 𝐵 ) )