Theorem fo2nd 6365

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 2818	. . . . . 6 |- x e. _V
2	1	snex 4303	. . . . 5 |- { x } e. _V
3	2	rnex 5030	. . . 4 |- ran { x } e. _V
4	3	uniex 4563	. . 3 |- U. ran { x } e. _V
5		df-2nd 6348	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5492	. 2 |- 2nd Fn _V
7	5	rnmpt 5010	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2818	. . . . 5 |- y e. _V
9	8, 8	opex 4350	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5252	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2238	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3705	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4991	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3930	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2246	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2923	. . . . . 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 2349	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2258	. 2 |- ran 2nd = _V
21		df-fo 5363	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 951	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1398   e. wcel 2205   {cab 2220   E.wrex 2523   _Vcvv 2815   {csn 3694   <.cop 3697   U.cuni 3919   ran crn 4755   Fn wfn 5352   -onto->wfo 5355   2ndc2nd 6346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-fo 5363  df-2nd 6348

This theorem is referenced by:  2ndcof  6371  2ndexg  6375  df2nd2  6429  2ndconst  6431  suplocexprlemmu  8049  suplocexprlemdisj  8051  suplocexprlemloc  8052  suplocexprlemub  8054  upxp  15263  uptx  15265  cnmpt2nd  15280