Theorem fo2nd 6056

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 2689	. . . . . 6 |- x e. _V
2	1	snex 4109	. . . . 5 |- { x } e. _V
3	2	rnex 4806	. . . 4 |- ran { x } e. _V
4	3	uniex 4359	. . 3 |- U. ran { x } e. _V
5		df-2nd 6039	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5251	. 2 |- 2nd Fn _V
7	5	rnmpt 4787	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2689	. . . . 5 |- y e. _V
9	8, 8	opex 4151	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5023	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2143	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3538	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4768	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3747	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2151	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2789	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. ran { <. y , y >. } ) -> E. x e. _V y = U. ran { x } )
17	9, 11, 16	mp2an 422	. . . . 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 2254	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2163	. 2 |- ran 2nd = _V
21		df-fo 5129	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 926	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1331   e. wcel 1480   {cab 2125   E.wrex 2417   _Vcvv 2686   {csn 3527   <.cop 3530   U.cuni 3736   ran crn 4540   Fn wfn 5118   -onto->wfo 5121   2ndc2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-fo 5129  df-2nd 6039

This theorem is referenced by:  2ndcof  6062  2ndexg  6066  df2nd2  6117  2ndconst  6119  suplocexprlemmu  7526  suplocexprlemdisj  7528  suplocexprlemloc  7529  suplocexprlemub  7531  upxp  12441  uptx  12443  cnmpt2nd  12458