Theorem fo2nd 6320

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 2805	. . . . . 6 |- x e. _V
2	1	snex 4275	. . . . 5 |- { x } e. _V
3	2	rnex 5000	. . . 4 |- ran { x } e. _V
4	3	uniex 4534	. . 3 |- U. ran { x } e. _V
5		df-2nd 6303	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5461	. 2 |- 2nd Fn _V
7	5	rnmpt 4980	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2805	. . . . 5 |- y e. _V
9	8, 8	opex 4321	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5221	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2235	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3680	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4961	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3904	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2243	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2910	. . . . . 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 2346	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2255	. 2 |- ran 2nd = _V
21		df-fo 5332	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 950	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   {csn 3669   <.cop 3672   U.cuni 3893   ran crn 4726   Fn wfn 5321   -onto->wfo 5324   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-fo 5332  df-2nd 6303

This theorem is referenced by:  2ndcof  6326  2ndexg  6330  df2nd2  6384  2ndconst  6386  suplocexprlemmu  7937  suplocexprlemdisj  7939  suplocexprlemloc  7940  suplocexprlemub  7942  upxp  14995  uptx  14997  cnmpt2nd  15012