Theorem cnvexg 4875

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 4723	. . 3 |- Rel `' A
2		relssdmrn 4861	. . 3 |- ( Rel `' A -> `' A C_ ( dom `' A X. ran `' A ) )
3	1, 2	ax-mp 7	. 2 |- `' A C_ ( dom `' A X. ran `' A )
4		df-rn 4374	. . . 4 |- ran A = dom `' A
5		rnexg 4615	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	syl5eqelr 2166	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4545	. . . 4 |- dom A = ran `' A
8		dmexg 4614	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	syl5eqelr 2166	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4470	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 403	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 3917	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 405	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 1433   _Vcvv 2601   C_ wss 2973   X. cxp 4361   `'ccnv 4362   dom cdm 4363   ran crn 4364   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  cnvex  4876  relcnvexb  4877  cofunex2g  5759  cnvf1o  5866  brtpos2  5889  tposexg  5896  cnven  6311  fopwdom  6333  relcnvfi  6391