Theorem cnvexg 5266

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 5106	. . 3 |- Rel `' A
2		relssdmrn 5249	. . 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 4730	. . . 4 |- ran A = dom `' A
5		rnexg 4989	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2317	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4915	. . . 4 |- dom A = ran `' A
8		dmexg 4988	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2317	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4833	. . 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 4223	. 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 2799   C_ wss 3197   X. cxp 4717   `'ccnv 4718   dom cdm 4719   ran crn 4720   Rel wrel 4724

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  cnvex  5267  relcnvexb  5268  cofunex2g  6255  cnvf1o  6371  brtpos2  6397  tposexg  6404  cnven  6961  cnvct  6962  fopwdom  6997  relcnvfi  7108  ennnfonelemim  12995  xpsval  13385  isunitd  14070  znval  14600  znle  14601  znbaslemnn  14603  znleval  14617  pw1nct  16369