Theorem isof1oopb 6575

Description: A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.)

Assertion

Ref	Expression
isof1oopb	|- ( H : A -1-1-onto-> B <-> H Isom ( _V X. _V ) , ( _V X. _V ) ( A , B ) )

Proof of Theorem isof1oopb

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

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . . . . 9 |- ( H ` x ) e. _V
2		fvex 6201	. . . . . . . . 9 |- ( H ` y ) e. _V
3	1, 2	opelvv 5166	. . . . . . . 8 |- <. ( H ` x ) , ( H ` y ) >. e. ( _V X. _V )
4		df-br 4654	. . . . . . . 8 |- ( ( H ` x ) ( _V X. _V ) ( H ` y ) <-> <. ( H ` x ) , ( H ` y ) >. e. ( _V X. _V ) )
5	3, 4	mpbir 221	. . . . . . 7 |- ( H ` x ) ( _V X. _V ) ( H ` y )
6	5	a1i 11	. . . . . 6 |- ( x ( _V X. _V ) y -> ( H ` x ) ( _V X. _V ) ( H ` y ) )
7		opelvvg 5165	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> <. x , y >. e. ( _V X. _V ) )
8		df-br 4654	. . . . . . . 8 |- ( x ( _V X. _V ) y <-> <. x , y >. e. ( _V X. _V ) )
9	7, 8	sylibr 224	. . . . . . 7 |- ( ( x e. A /\ y e. A ) -> x ( _V X. _V ) y )
10	9	a1d 25	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( ( H ` x ) ( _V X. _V ) ( H ` y ) -> x ( _V X. _V ) y ) )
11	6, 10	impbid2 216	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( x ( _V X. _V ) y <-> ( H ` x ) ( _V X. _V ) ( H ` y ) ) )
12	11	adantl 482	. . . 4 |- ( ( H : A -1-1-onto-> B /\ ( x e. A /\ y e. A ) ) -> ( x ( _V X. _V ) y <-> ( H ` x ) ( _V X. _V ) ( H ` y ) ) )
13	12	ralrimivva 2971	. . 3 |- ( H : A -1-1-onto-> B -> A. x e. A A. y e. A ( x ( _V X. _V ) y <-> ( H ` x ) ( _V X. _V ) ( H ` y ) ) )
14	13	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 ( _V X. _V ) y <-> ( H ` x ) ( _V X. _V ) ( H ` y ) ) ) )
15		df-isom 5897	. 2 |- ( H Isom ( _V X. _V ) , ( _V X. _V ) ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x ( _V X. _V ) y <-> ( H ` x ) ( _V X. _V ) ( H ` y ) ) ) )
16	14, 15	bitr4i 267	1 |- ( H : A -1-1-onto-> B <-> H Isom ( _V X. _V ) , ( _V X. _V ) ( A , B ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   -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-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-xp 5120  df-iota 5851  df-fv 5896  df-isom 5897

This theorem is referenced by: (None)