Theorem cnvexg 4963

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 4805	. . 3 |- Rel `' A
2		relssdmrn 4946	. . 3 |- ( Rel `' A -> `' A C_ ( dom `' A X. ran `' A ) )
3	1, 2	ax-mp 7	. 2 |- `' A C_ ( dom `' A X. ran `' A )
4		df-rn 4447	. . . 4 |- ran A = dom `' A
5		rnexg 4694	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	syl5eqelr 2175	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4624	. . . 4 |- dom A = ran `' A
8		dmexg 4693	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	syl5eqelr 2175	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4548	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 403	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 3976	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 405	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 1438   _Vcvv 2619   C_ wss 2999   X. cxp 4434   `'ccnv 4435   dom cdm 4436   ran crn 4437   Rel wrel 4441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-dm 4446  df-rn 4447

This theorem is referenced by:  cnvex  4964  relcnvexb  4965  cofunex2g  5875  cnvf1o  5982  brtpos2  6008  tposexg  6015  cnven  6515  cnvct  6516  fopwdom  6542  relcnvfi  6640