Theorem cnvexg 5203

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 5043	. . 3 |- Rel `' A
2		relssdmrn 5186	. . 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 4670	. . . 4 |- ran A = dom `' A
5		rnexg 4927	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2281	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4854	. . . 4 |- dom A = ran `' A
8		dmexg 4926	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2281	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4773	. . 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 4168	. 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 2164   _Vcvv 2760   C_ wss 3153   X. cxp 4657   `'ccnv 4658   dom cdm 4659   ran crn 4660   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  cnvex  5204  relcnvexb  5205  cofunex2g  6162  cnvf1o  6278  brtpos2  6304  tposexg  6311  cnven  6862  cnvct  6863  fopwdom  6892  relcnvfi  7000  ennnfonelemim  12581  xpsval  12935  isunitd  13602  znval  14124  znle  14125  znbaslemnn  14127  znleval  14141  pw1nct  15493