Theorem fo2nd 6243

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 2774	. . . . . 6 |- x e. _V
2	1	snex 4228	. . . . 5 |- { x } e. _V
3	2	rnex 4945	. . . 4 |- ran { x } e. _V
4	3	uniex 4483	. . 3 |- U. ran { x } e. _V
5		df-2nd 6226	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5403	. 2 |- 2nd Fn _V
7	5	rnmpt 4925	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2774	. . . . 5 |- y e. _V
9	8, 8	opex 4272	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5166	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2208	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3643	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4906	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3860	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2216	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2876	. . . . . 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 2319	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2228	. 2 |- ran 2nd = _V
21		df-fo 5276	. 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 1372   e. wcel 2175   {cab 2190   E.wrex 2484   _Vcvv 2771   {csn 3632   <.cop 3635   U.cuni 3849   ran crn 4675   Fn wfn 5265   -onto->wfo 5268   2ndc2nd 6224

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273  df-fo 5276  df-2nd 6226

This theorem is referenced by:  2ndcof  6249  2ndexg  6253  df2nd2  6305  2ndconst  6307  suplocexprlemmu  7830  suplocexprlemdisj  7832  suplocexprlemloc  7833  suplocexprlemub  7835  upxp  14715  uptx  14717  cnmpt2nd  14732