Theorem cnvexg 5320

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 5160	. . 3 |- Rel `' A
2		relssdmrn 5303	. . 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 4780	. . . 4 |- ran A = dom `' A
5		rnexg 5042	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2326	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4968	. . . 4 |- dom A = ran `' A
8		dmexg 5041	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2326	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4884	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 415	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 4267	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 418	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   C_ wss 3220   X. cxp 4767   `'ccnv 4768   dom cdm 4769   ran crn 4770   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  cnvex  5321  relcnvexb  5322  cofunex2g  6329  cnvf1o  6451  brtpos2  6512  tposexg  6519  cnven  7086  cnvct  7087  fopwdom  7126  relcnvfi  7245  ennnfonelemim  13293  xpsval  14178  isunitd  14386  znval  14943  znle  14944  znbaslemnn  14946  znleval  14960  pw1nct  16947