Theorem cnvexg 5141

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 4982	. . 3 |- Rel `' A
2		relssdmrn 5124	. . 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 4615	. . . 4 |- ran A = dom `' A
5		rnexg 4869	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2254	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4796	. . . 4 |- dom A = ran `' A
8		dmexg 4868	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2254	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4718	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 409	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 4121	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 411	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 2136   _Vcvv 2726   C_ wss 3116   X. cxp 4602   `'ccnv 4603   dom cdm 4604   ran crn 4605   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  cnvex  5142  relcnvexb  5143  cofunex2g  6078  cnvf1o  6193  brtpos2  6219  tposexg  6226  cnven  6774  cnvct  6775  fopwdom  6802  relcnvfi  6906  ennnfonelemim  12357  pw1nct  13893