Theorem isof1oidb 6574

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

Assertion

Ref	Expression
isof1oidb	|- ( H : A -1-1-onto-> B <-> H Isom _I , _I ( A , B ) )

Proof of Theorem isof1oidb

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1of1 6136	. . . . . 6 |- ( H : A -1-1-onto-> B -> H : A -1-1-> B )
2		f1fveq 6519	. . . . . 6 |- ( ( H : A -1-1-> B /\ ( x e. A /\ y e. A ) ) -> ( ( H ` x ) = ( H ` y ) <-> x = y ) )
3	1, 2	sylan 488	. . . . 5 |- ( ( H : A -1-1-onto-> B /\ ( x e. A /\ y e. A ) ) -> ( ( H ` x ) = ( H ` y ) <-> x = y ) )
4		fvex 6201	. . . . . . 7 |- ( H ` y ) e. _V
5	4	ideq 5274	. . . . . 6 |- ( ( H ` x ) _I ( H ` y ) <-> ( H ` x ) = ( H ` y ) )
6	5	a1i 11	. . . . 5 |- ( ( H : A -1-1-onto-> B /\ ( x e. A /\ y e. A ) ) -> ( ( H ` x ) _I ( H ` y ) <-> ( H ` x ) = ( H ` y ) ) )
7		ideqg 5273	. . . . . 6 |- ( y e. A -> ( x _I y <-> x = y ) )
8	7	ad2antll 765	. . . . 5 |- ( ( H : A -1-1-onto-> B /\ ( x e. A /\ y e. A ) ) -> ( x _I y <-> x = y ) )
9	3, 6, 8	3bitr4rd 301	. . . 4 |- ( ( H : A -1-1-onto-> B /\ ( x e. A /\ y e. A ) ) -> ( x _I y <-> ( H ` x ) _I ( H ` y ) ) )
10	9	ralrimivva 2971	. . 3 |- ( H : A -1-1-onto-> B -> A. x e. A A. y e. A ( x _I y <-> ( H ` x ) _I ( H ` y ) ) )
11	10	pm4.71i 664	. 2 |- ( H : A -1-1-onto-> B <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x _I y <-> ( H ` x ) _I ( H ` y ) ) ) )
12		df-isom 5897	. 2 |- ( H Isom _I , _I ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x _I y <-> ( H ` x ) _I ( H ` y ) ) ) )
13	11, 12	bitr4i 267	1 |- ( H : A -1-1-onto-> B <-> H Isom _I , _I ( A , B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   _I cid 5023   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by: (None)