Theorem cnvexg 5328

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 5168	. . 3 |- Rel `' A
2		relssdmrn 5311	. . 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 4787	. . . 4 |- ran A = dom `' A
5		rnexg 5049	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2326	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4975	. . . 4 |- dom A = ran `' A
8		dmexg 5048	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2326	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4891	. . 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 4274	. 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 4774   `'ccnv 4775   dom cdm 4776   ran crn 4777   Rel wrel 4781

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

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 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787

This theorem is used by:  cnvex  5329  relcnvexb  5330  cofunex2g  6343  cnvf1o  6465  brtpos2  6526  tposexg  6533  cnven  7100  cnvct  7101  fopwdom  7140  relcnvfi  7259  ennnfonelemim  13373  xpsval  14308  isunitd  14522  znval  15080  znle  15081  znbaslemnn  15083  znleval  15097  pw1nct  17238