Theorem cnvf1o 6389

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 2231	. 2 |- ( x e. A |-> U. `' { x } ) = ( x e. A |-> U. `' { x } )
2		snexg 4274	. . . 4 |- ( x e. A -> { x } e. _V )
3		cnvexg 5274	. . . 4 |- ( { x } e. _V -> `' { x } e. _V )
4		uniexg 4536	. . . 4 |- ( `' { x } e. _V -> U. `' { x } e. _V )
5	2, 3, 4	3syl 17	. . 3 |- ( x e. A -> U. `' { x } e. _V )
6	5	adantl 277	. 2 |- ( ( Rel A /\ x e. A ) -> U. `' { x } e. _V )
7		snexg 4274	. . . 4 |- ( y e. `' A -> { y } e. _V )
8		cnvexg 5274	. . . 4 |- ( { y } e. _V -> `' { y } e. _V )
9		uniexg 4536	. . . 4 |- ( `' { y } e. _V -> U. `' { y } e. _V )
10	7, 8, 9	3syl 17	. . 3 |- ( y e. `' A -> U. `' { y } e. _V )
11	10	adantl 277	. 2 |- ( ( Rel A /\ y e. `' A ) -> U. `' { y } e. _V )
12		cnvf1olem 6388	. . 3 |- ( ( Rel A /\ ( x e. A /\ y = U. `' { x } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
13		relcnv 5114	. . . . 5 |- Rel `' A
14		simpr 110	. . . . 5 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
15		cnvf1olem 6388	. . . . 5 |- ( ( Rel `' A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. `' `' A /\ y = U. `' { x } ) )
16	13, 14, 15	sylancr 414	. . . 4 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. `' `' A /\ y = U. `' { x } ) )
17		dfrel2 5187	. . . . . . 7 |- ( Rel A <-> `' `' A = A )
18		eleq2 2295	. . . . . . 7 |- ( `' `' A = A -> ( x e. `' `' A <-> x e. A ) )
19	17, 18	sylbi 121	. . . . . 6 |- ( Rel A -> ( x e. `' `' A <-> x e. A ) )
20	19	anbi1d 465	. . . . 5 |- ( Rel A -> ( ( x e. `' `' A /\ y = U. `' { x } ) <-> ( x e. A /\ y = U. `' { x } ) ) )
21	20	adantr 276	. . . 4 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( ( x e. `' `' A /\ y = U. `' { x } ) <-> ( x e. A /\ y = U. `' { x } ) ) )
22	16, 21	mpbid 147	. . 3 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( x e. A /\ y = U. `' { x } ) )
23	12, 22	impbida 600	. 2 |- ( Rel A -> ( ( x e. A /\ y = U. `' { x } ) <-> ( y e. `' A /\ x = U. `' { y } ) ) )
24	1, 6, 11, 23	f1od 6225	1 |- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   U.cuni 3893   |-> cmpt 4150   `'ccnv 4724   Rel wrel 4730   -1-1-onto->wf1o 5325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  tposf12  6434  cnven  6982  xpcomf1o  7008  fsumcnv  11997  fprodcnv  12185