Theorem cnvf1o 7276

Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
cnvf1o	|- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )

Distinct variable group:   x, A

Proof of Theorem cnvf1o

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( x e. A |-> U. `' { x } ) = ( x e. A |-> U. `' { x } )
2		snex 4908	. . . . 5 |- { x } e. _V
3	2	cnvex 7113	. . . 4 |- `' { x } e. _V
4	3	uniex 6953	. . 3 |- U. `' { x } e. _V
5	4	a1i 11	. 2 |- ( ( Rel A /\ x e. A ) -> U. `' { x } e. _V )
6		snex 4908	. . . . 5 |- { y } e. _V
7	6	cnvex 7113	. . . 4 |- `' { y } e. _V
8	7	uniex 6953	. . 3 |- U. `' { y } e. _V
9	8	a1i 11	. 2 |- ( ( Rel A /\ y e. `' A ) -> U. `' { y } e. _V )
10		cnvf1olem 7275	. . 3 |- ( ( Rel A /\ ( x e. A /\ y = U. `' { x } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
11		relcnv 5503	. . . . 5 |- Rel `' A
12		simpr 477	. . . . 5 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
13		cnvf1olem 7275	. . . . 5 |- ( ( Rel `' A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. `' `' A /\ y = U. `' { x } ) )
14	11, 12, 13	sylancr 695	. . . 4 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. `' `' A /\ y = U. `' { x } ) )
15		dfrel2 5583	. . . . . . 7 |- ( Rel A <-> `' `' A = A )
16		eleq2 2690	. . . . . . 7 |- ( `' `' A = A -> ( x e. `' `' A <-> x e. A ) )
17	15, 16	sylbi 207	. . . . . 6 |- ( Rel A -> ( x e. `' `' A <-> x e. A ) )
18	17	anbi1d 741	. . . . 5 |- ( Rel A -> ( ( x e. `' `' A /\ y = U. `' { x } ) <-> ( x e. A /\ y = U. `' { x } ) ) )
19	18	adantr 481	. . . 4 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( ( x e. `' `' A /\ y = U. `' { x } ) <-> ( x e. A /\ y = U. `' { x } ) ) )
20	14, 19	mpbid 222	. . 3 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. A /\ y = U. `' { x } ) )
21	10, 20	impbida 877	. 2 |- ( Rel A -> ( ( x e. A /\ y = U. `' { x } ) <-> ( y e. `' A /\ x = U. `' { y } ) ) )
22	1, 5, 9, 21	f1od 6885	1 |- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   Rel wrel 5119   -1-1-onto->wf1o 5887

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  tposf12  7377  cnven  8032  xpcomf1o  8049  fsumcnv  14504  fprodcnv  14713  gsumcom2  18374