Theorem cnvexg 5158

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 4999	. . 3 |- Rel `' A
2		relssdmrn 5141	. . 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 4631	. . . 4 |- ran A = dom `' A
5		rnexg 4885	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2263	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4812	. . . 4 |- dom A = ran `' A
8		dmexg 4884	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2263	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4734	. . 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 4137	. 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 2146   _Vcvv 2735   C_ wss 3127   X. cxp 4618   `'ccnv 4619   dom cdm 4620   ran crn 4621   Rel wrel 4625

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by:  cnvex  5159  relcnvexb  5160  cofunex2g  6101  cnvf1o  6216  brtpos2  6242  tposexg  6249  cnven  6798  cnvct  6799  fopwdom  6826  relcnvfi  6930  ennnfonelemim  12392  pw1nct  14313