Theorem cnvexg 5281

Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Assertion

Ref	Expression
cnvexg	|- ( A e. V -> `' A e. _V )

Proof of Theorem cnvexg

Step	Hyp	Ref	Expression
1		relcnv 5121	. . 3 |- Rel `' A
2		relssdmrn 5264	. . 3 |- ( Rel `' A -> `' A C_ ( dom `' A X. ran `' A ) )
3	1, 2	ax-mp 5	. 2 |- `' A C_ ( dom `' A X. ran `' A )
4		df-rn 4742	. . . 4 |- ran A = dom `' A
5		rnexg 5003	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2319	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4929	. . . 4 |- dom A = ran `' A
8		dmexg 5002	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2319	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4846	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 411	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 4233	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 414	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2803   C_ wss 3201   X. cxp 4729   `'ccnv 4730   dom cdm 4731   ran crn 4732   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  cnvex  5282  relcnvexb  5283  cofunex2g  6281  cnvf1o  6399  brtpos2  6460  tposexg  6467  cnven  7026  cnvct  7027  fopwdom  7065  relcnvfi  7183  ennnfonelemim  13125  xpsval  13515  isunitd  14201  znval  14732  znle  14733  znbaslemnn  14735  znleval  14749  pw1nct  16725