Theorem cnvexg 5325

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 5165	. . 3 |- Rel `' A
2		relssdmrn 5308	. . 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 4785	. . . 4 |- ran A = dom `' A
5		rnexg 5047	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2326	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4973	. . . 4 |- dom A = ran `' A
8		dmexg 5046	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2326	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4889	. . 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 4272	. 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 )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   C_ wss 3220   X. cxp 4772   `'ccnv 4773   dom cdm 4774   ran crn 4775   Rel wrel 4779

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  cnvex  5326  relcnvexb  5327  cofunex2g  6339  cnvf1o  6461  brtpos2  6522  tposexg  6529  cnven  7096  cnvct  7097  fopwdom  7136  relcnvfi  7255  ennnfonelemim  13315  xpsval  14201  isunitd  14413  znval  14971  znle  14972  znbaslemnn  14974  znleval  14988  pw1nct  17033