Theorem fo2nd 6161

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 2742	. . . . . 6 |- x e. _V
2	1	snex 4187	. . . . 5 |- { x } e. _V
3	2	rnex 4896	. . . 4 |- ran { x } e. _V
4	3	uniex 4439	. . 3 |- U. ran { x } e. _V
5		df-2nd 6144	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5346	. 2 |- 2nd Fn _V
7	5	rnmpt 4877	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2742	. . . . 5 |- y e. _V
9	8, 8	opex 4231	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5115	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2181	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3605	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4858	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3822	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2189	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2843	. . . . . 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 2292	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2201	. 2 |- ran 2nd = _V
21		df-fo 5224	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 942	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2739   {csn 3594   <.cop 3597   U.cuni 3811   ran crn 4629   Fn wfn 5213   -onto->wfo 5216   2ndc2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-fo 5224  df-2nd 6144

This theorem is referenced by:  2ndcof  6167  2ndexg  6171  df2nd2  6223  2ndconst  6225  suplocexprlemmu  7719  suplocexprlemdisj  7721  suplocexprlemloc  7722  suplocexprlemub  7724  upxp  13857  uptx  13859  cnmpt2nd  13874