Theorem cnvexg 5204

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 5044	. . 3 |- Rel `' A
2		relssdmrn 5187	. . 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 4671	. . . 4 |- ran A = dom `' A
5		rnexg 4928	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2281	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4855	. . . 4 |- dom A = ran `' A
8		dmexg 4927	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2281	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4774	. . 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 4169	. 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 2164   _Vcvv 2760   C_ wss 3154   X. cxp 4658   `'ccnv 4659   dom cdm 4660   ran crn 4661   Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  cnvex  5205  relcnvexb  5206  cofunex2g  6164  cnvf1o  6280  brtpos2  6306  tposexg  6313  cnven  6864  cnvct  6865  fopwdom  6894  relcnvfi  7002  ennnfonelemim  12584  xpsval  12938  isunitd  13605  znval  14135  znle  14136  znbaslemnn  14138  znleval  14152  pw1nct  15563