Theorem fo2nd 6126

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 2729	. . . . . 6 |- x e. _V
2	1	snex 4164	. . . . 5 |- { x } e. _V
3	2	rnex 4871	. . . 4 |- ran { x } e. _V
4	3	uniex 4415	. . 3 |- U. ran { x } e. _V
5		df-2nd 6109	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5316	. 2 |- 2nd Fn _V
7	5	rnmpt 4852	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2729	. . . . 5 |- y e. _V
9	8, 8	opex 4207	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5088	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2169	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3587	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4833	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3800	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2177	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2830	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. ran { <. y , y >. } ) -> E. x e. _V y = U. ran { x } )
17	9, 11, 16	mp2an 423	. . . . 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 2281	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2189	. 2 |- ran 2nd = _V
21		df-fo 5194	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 932	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726   {csn 3576   <.cop 3579   U.cuni 3789   ran crn 4605   Fn wfn 5183   -onto->wfo 5186   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-fo 5194  df-2nd 6109

This theorem is referenced by:  2ndcof  6132  2ndexg  6136  df2nd2  6188  2ndconst  6190  suplocexprlemmu  7659  suplocexprlemdisj  7661  suplocexprlemloc  7662  suplocexprlemub  7664  upxp  12912  uptx  12914  cnmpt2nd  12929