Theorem cnvexg 5274

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 5114	. . 3 |- Rel `' A
2		relssdmrn 5257	. . 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 4736	. . . 4 |- ran A = dom `' A
5		rnexg 4997	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2319	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4923	. . . 4 |- dom A = ran `' A
8		dmexg 4996	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2319	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4840	. . 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 4228	. 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 2202   _Vcvv 2802   C_ wss 3200   X. cxp 4723   `'ccnv 4724   dom cdm 4725   ran crn 4726   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  cnvex  5275  relcnvexb  5276  cofunex2g  6271  cnvf1o  6389  brtpos2  6416  tposexg  6423  cnven  6982  cnvct  6983  fopwdom  7021  relcnvfi  7139  ennnfonelemim  13044  xpsval  13434  isunitd  14119  znval  14649  znle  14650  znbaslemnn  14652  znleval  14666  pw1nct  16604