Theorem cnvexg 5300

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 5140	. . 3 |- Rel `' A
2		relssdmrn 5283	. . 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 4760	. . . 4 |- ran A = dom `' A
5		rnexg 5022	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2320	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4948	. . . 4 |- dom A = ran `' A
8		dmexg 5021	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2320	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4864	. . 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 4249	. 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 2203   _Vcvv 2813   C_ wss 3211   X. cxp 4747   `'ccnv 4748   dom cdm 4749   ran crn 4750   Rel wrel 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  cnvex  5301  relcnvexb  5302  cofunex2g  6303  cnvf1o  6421  brtpos2  6482  tposexg  6489  cnven  7049  cnvct  7050  fopwdom  7089  relcnvfi  7208  ennnfonelemim  13175  xpsval  13565  isunitd  14251  znval  14784  znle  14785  znbaslemnn  14787  znleval  14801  pw1nct  16777