Theorem cnvexg 5239

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 5079	. . 3 |- Rel `' A
2		relssdmrn 5222	. . 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 4704	. . . 4 |- ran A = dom `' A
5		rnexg 4962	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2295	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4889	. . . 4 |- dom A = ran `' A
8		dmexg 4961	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2295	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4807	. . 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 4199	. 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 2178   _Vcvv 2776   C_ wss 3174   X. cxp 4691   `'ccnv 4692   dom cdm 4693   ran crn 4694   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  cnvex  5240  relcnvexb  5241  cofunex2g  6218  cnvf1o  6334  brtpos2  6360  tposexg  6367  cnven  6924  cnvct  6925  fopwdom  6958  relcnvfi  7069  ennnfonelemim  12910  xpsval  13299  isunitd  13983  znval  14513  znle  14514  znbaslemnn  14516  znleval  14530  pw1nct  16142