Theorem fo2nd 6184

Description: The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fo2nd	|- 2nd : _V -onto-> _V

Proof of Theorem fo2nd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2755	. . . . . 6 |- x e. _V
2	1	snex 4203	. . . . 5 |- { x } e. _V
3	2	rnex 4912	. . . 4 |- ran { x } e. _V
4	3	uniex 4455	. . 3 |- U. ran { x } e. _V
5		df-2nd 6167	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5363	. 2 |- 2nd Fn _V
7	5	rnmpt 4893	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2755	. . . . 5 |- y e. _V
9	8, 8	opex 4247	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5131	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2193	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3618	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4874	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3835	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2201	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2856	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. ran { <. y , y >. } ) -> E. x e. _V y = U. ran { x } )
17	9, 11, 16	mp2an 426	. . . . 5 |- E. x e. _V y = U. ran { x }
18	8, 17	2th 174	. . . 4 |- ( y e. _V <-> E. x e. _V y = U. ran { x } )
19	18	abbi2i 2304	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2213	. 2 |- ran 2nd = _V
21		df-fo 5241	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 944	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1364   e. wcel 2160   {cab 2175   E.wrex 2469   _Vcvv 2752   {csn 3607   <.cop 3610   U.cuni 3824   ran crn 4645   Fn wfn 5230   -onto->wfo 5233   2ndc2nd 6165

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-fo 5241  df-2nd 6167

This theorem is referenced by:  2ndcof  6190  2ndexg  6194  df2nd2  6246  2ndconst  6248  suplocexprlemmu  7748  suplocexprlemdisj  7750  suplocexprlemloc  7751  suplocexprlemub  7753  upxp  14249  uptx  14251  cnmpt2nd  14266