Theorem fo2nd 6244

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 2775	. . . . . 6 |- x e. _V
2	1	snex 4229	. . . . 5 |- { x } e. _V
3	2	rnex 4946	. . . 4 |- ran { x } e. _V
4	3	uniex 4484	. . 3 |- U. ran { x } e. _V
5		df-2nd 6227	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5404	. 2 |- 2nd Fn _V
7	5	rnmpt 4926	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2775	. . . . 5 |- y e. _V
9	8, 8	opex 4273	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5167	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2209	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3644	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4907	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3861	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2217	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2877	. . . . . 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 2320	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2229	. 2 |- ran 2nd = _V
21		df-fo 5277	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 945	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   {csn 3633   <.cop 3636   U.cuni 3850   ran crn 4676   Fn wfn 5266   -onto->wfo 5269   2ndc2nd 6225

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274  df-fo 5277  df-2nd 6227

This theorem is referenced by:  2ndcof  6250  2ndexg  6254  df2nd2  6306  2ndconst  6308  suplocexprlemmu  7831  suplocexprlemdisj  7833  suplocexprlemloc  7834  suplocexprlemub  7836  upxp  14744  uptx  14746  cnmpt2nd  14761