Theorem cnvexg 5076

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 4917	. . 3 |- Rel `' A
2		relssdmrn 5059	. . 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 4550	. . . 4 |- ran A = dom `' A
5		rnexg 4804	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2227	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4731	. . . 4 |- dom A = ran `' A
8		dmexg 4803	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2227	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4653	. . 3 |- ( ( dom `' A e. _V /\ ran `' A e. _V ) -> ( dom `' A X. ran `' A ) e. _V )
11	6, 9, 10	syl2anc 408	. 2 |- ( A e. V -> ( dom `' A X. ran `' A ) e. _V )
12		ssexg 4067	. 2 |- ( ( `' A C_ ( dom `' A X. ran `' A ) /\ ( dom `' A X. ran `' A ) e. _V ) -> `' A e. _V )
13	3, 11, 12	sylancr 410	1 |- ( A e. V -> `' A e. _V )

Syntax hints:   -> wi 4   e. wcel 1480   _Vcvv 2686   C_ wss 3071   X. cxp 4537   `'ccnv 4538   dom cdm 4539   ran crn 4540   Rel wrel 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by:  cnvex  5077  relcnvexb  5078  cofunex2g  6010  cnvf1o  6122  brtpos2  6148  tposexg  6155  cnven  6702  cnvct  6703  fopwdom  6730  relcnvfi  6829  ennnfonelemim  11937