Theorem cnven 6982

Description: A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
cnven	|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )

Proof of Theorem cnven

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. 2 |- ( ( Rel A /\ A e. V ) -> A e. V )
2		cnvexg 5274	. . 3 |- ( A e. V -> `' A e. _V )
3	2	adantl 277	. 2 |- ( ( Rel A /\ A e. V ) -> `' A e. _V )
4		cnvf1o 6389	. . 3 |- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )
5	4	adantr 276	. 2 |- ( ( Rel A /\ A e. V ) -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )
6		f1oen2g 6927	. 2 |- ( ( A e. V /\ `' A e. _V /\ ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A ) -> A ~~ `' A )
7	1, 3, 5, 6	syl3anc 1273	1 |- ( ( Rel A /\ A e. V ) -> A ~~ `' A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2802   {csn 3669   U.cuni 3893   class class class wbr 4088   |-> cmpt 4150   `'ccnv 4724   Rel wrel 4730   -1-1-onto->wf1o 5325   ~~ cen 6906

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  df-en 6909

This theorem is referenced by:  cnvct  6983  relcnvfi  7139  lgsquadlem3  15807