Theorem cnvf1o 6371

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 2229	. 2 |- ( x e. A |-> U. `' { x } ) = ( x e. A |-> U. `' { x } )
2		snexg 4268	. . . 4 |- ( x e. A -> { x } e. _V )
3		cnvexg 5266	. . . 4 |- ( { x } e. _V -> `' { x } e. _V )
4		uniexg 4530	. . . 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 4268	. . . 4 |- ( y e. `' A -> { y } e. _V )
8		cnvexg 5266	. . . 4 |- ( { y } e. _V -> `' { y } e. _V )
9		uniexg 4530	. . . 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 6370	. . 3 |- ( ( Rel A /\ ( x e. A /\ y = U. `' { x } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
13		relcnv 5106	. . . . 5 |- Rel `' A
14		simpr 110	. . . . 5 |- ( ( Rel A /\ ( y e. `' A /\ x = U. `' { y } ) ) -> ( y e. `' A /\ x = U. `' { y } ) )
15		cnvf1olem 6370	. . . . 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 5179	. . . . . . 7 |- ( Rel A <-> `' `' A = A )
18		eleq2 2293	. . . . . . 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 598	. 2 |- ( Rel A -> ( ( x e. A /\ y = U. `' { x } ) <-> ( y e. `' A /\ x = U. `' { y } ) ) )
24	1, 6, 11, 23	f1od 6209	1 |- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   U.cuni 3888   |-> cmpt 4145   `'ccnv 4718   Rel wrel 4724   -1-1-onto->wf1o 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287

This theorem is referenced by:  tposf12  6415  cnven  6961  xpcomf1o  6984  fsumcnv  11948  fprodcnv  12136