Theorem cnvexg 4855

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 4703	. . 3 |- Rel `' A
2		relssdmrn 4841	. . 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 4356	. . . 4 |- ran A = dom `' A
5		rnexg 4597	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	syl5eqelr 2125	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4527	. . . 4 |- dom A = ran `' A
8		dmexg 4596	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	syl5eqelr 2125	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4452	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 391	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 3896	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 393	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 1393   _Vcvv 2557   C_ wss 2917   X. cxp 4343   `'ccnv 4344   dom cdm 4345   ran crn 4346   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  cnvex  4856  relcnvexb  4857  cofunex2g  5739  cnvf1o  5846  brtpos2  5866  tposexg  5873  cnven  6288  fopwdom  6310