Theorem fo2nd 6302

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 2802	. . . . . 6 |- x e. _V
2	1	snex 4268	. . . . 5 |- { x } e. _V
3	2	rnex 4991	. . . 4 |- ran { x } e. _V
4	3	uniex 4527	. . 3 |- U. ran { x } e. _V
5		df-2nd 6285	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5451	. 2 |- 2nd Fn _V
7	5	rnmpt 4971	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2802	. . . . 5 |- y e. _V
9	8, 8	opex 4314	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5212	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2233	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3677	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4952	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3898	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2241	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2907	. . . . . 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 2344	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2253	. 2 |- ran 2nd = _V
21		df-fo 5323	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 948	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   {csn 3666   <.cop 3669   U.cuni 3887   ran crn 4719   Fn wfn 5312   -onto->wfo 5315   2ndc2nd 6283

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-fo 5323  df-2nd 6285

This theorem is referenced by:  2ndcof  6308  2ndexg  6312  df2nd2  6364  2ndconst  6366  suplocexprlemmu  7901  suplocexprlemdisj  7903  suplocexprlemloc  7904  suplocexprlemub  7906  upxp  14940  uptx  14942  cnmpt2nd  14957