Theorem cnvexg 5084

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 4925	. . 3 |- Rel `' A
2		relssdmrn 5067	. . 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 4558	. . . 4 |- ran A = dom `' A
5		rnexg 4812	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2228	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4739	. . . 4 |- dom A = ran `' A
8		dmexg 4811	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2228	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4661	. . 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 4075	. 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 1481   _Vcvv 2689   C_ wss 3076   X. cxp 4545   `'ccnv 4546   dom cdm 4547   ran crn 4548   Rel wrel 4552

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  cnvex  5085  relcnvexb  5086  cofunex2g  6018  cnvf1o  6130  brtpos2  6156  tposexg  6163  cnven  6710  cnvct  6711  fopwdom  6738  relcnvfi  6837  ennnfonelemim  11973  pw1nct  13371