Theorem cnvexg 5220

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 5060	. . 3 |- Rel `' A
2		relssdmrn 5203	. . 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 4686	. . . 4 |- ran A = dom `' A
5		rnexg 4943	. . . 4 |- ( A e. V -> ran A e. _V )
6	4, 5	eqeltrrid 2293	. . 3 |- ( A e. V -> dom `' A e. _V )
7		dfdm4 4870	. . . 4 |- dom A = ran `' A
8		dmexg 4942	. . . 4 |- ( A e. V -> dom A e. _V )
9	7, 8	eqeltrrid 2293	. . 3 |- ( A e. V -> ran `' A e. _V )
10		xpexg 4789	. . 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 4183	. 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 2176   _Vcvv 2772   C_ wss 3166   X. cxp 4673   `'ccnv 4674   dom cdm 4675   ran crn 4676   Rel wrel 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  cnvex  5221  relcnvexb  5222  cofunex2g  6195  cnvf1o  6311  brtpos2  6337  tposexg  6344  cnven  6900  cnvct  6901  fopwdom  6933  relcnvfi  7043  ennnfonelemim  12795  xpsval  13184  isunitd  13868  znval  14398  znle  14399  znbaslemnn  14401  znleval  14415  pw1nct  15940