Theorem cnvexg 5272

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 5112	. . 3 |- Rel `' A
2		relssdmrn 5255	. . 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 4734	. . . 4 |- ran A = dom `' A
5		rnexg 4995	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2317	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4921	. . . 4 |- dom A = ran `' A
8		dmexg 4994	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2317	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4838	. . 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 4226	. 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 2200   _Vcvv 2800   C_ wss 3198   X. cxp 4721   `'ccnv 4722   dom cdm 4723   ran crn 4724   Rel wrel 4728

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  cnvex  5273  relcnvexb  5274  cofunex2g  6267  cnvf1o  6385  brtpos2  6412  tposexg  6419  cnven  6978  cnvct  6979  fopwdom  7017  relcnvfi  7131  ennnfonelemim  13035  xpsval  13425  isunitd  14110  znval  14640  znle  14641  znbaslemnn  14643  znleval  14657  pw1nct  16540