Theorem fo2nd 6064

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 2692	. . . . . 6 |- x e. _V
2	1	snex 4117	. . . . 5 |- { x } e. _V
3	2	rnex 4814	. . . 4 |- ran { x } e. _V
4	3	uniex 4367	. . 3 |- U. ran { x } e. _V
5		df-2nd 6047	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5259	. 2 |- 2nd Fn _V
7	5	rnmpt 4795	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2692	. . . . 5 |- y e. _V
9	8, 8	opex 4159	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5031	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2144	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3543	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4776	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3755	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2152	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2793	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. ran { <. y , y >. } ) -> E. x e. _V y = U. ran { x } )
17	9, 11, 16	mp2an 423	. . . . 5 |- E. x e. _V y = U. ran { x }
18	8, 17	2th 173	. . . 4 |- ( y e. _V <-> E. x e. _V y = U. ran { x } )
19	18	abbi2i 2255	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2164	. 2 |- ran 2nd = _V
21		df-fo 5137	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 927	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   _Vcvv 2689   {csn 3532   <.cop 3535   U.cuni 3744   ran crn 4548   Fn wfn 5126   -onto->wfo 5129   2ndc2nd 6045

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-fo 5137  df-2nd 6047

This theorem is referenced by:  2ndcof  6070  2ndexg  6074  df2nd2  6125  2ndconst  6127  suplocexprlemmu  7550  suplocexprlemdisj  7552  suplocexprlemloc  7553  suplocexprlemub  7555  upxp  12480  uptx  12482  cnmpt2nd  12497