Theorem cnvexg 5167

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 5007	. . 3 |- Rel `' A
2		relssdmrn 5150	. . 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 4638	. . . 4 |- ran A = dom `' A
5		rnexg 4893	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2265	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4820	. . . 4 |- dom A = ran `' A
8		dmexg 4892	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2265	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4741	. . 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 4143	. 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 2148   _Vcvv 2738   C_ wss 3130   X. cxp 4625   `'ccnv 4626   dom cdm 4627   ran crn 4628   Rel wrel 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638

This theorem is referenced by:  cnvex  5168  relcnvexb  5169  cofunex2g  6111  cnvf1o  6226  brtpos2  6252  tposexg  6259  cnven  6808  cnvct  6809  fopwdom  6836  relcnvfi  6940  ennnfonelemim  12425  xpsval  12771  isunitd  13275  pw1nct  14755