Theorem fo2nd 6225

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 2766	. . . . . 6 |- x e. _V
2	1	snex 4219	. . . . 5 |- { x } e. _V
3	2	rnex 4934	. . . 4 |- ran { x } e. _V
4	3	uniex 4473	. . 3 |- U. ran { x } e. _V
5		df-2nd 6208	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5389	. 2 |- 2nd Fn _V
7	5	rnmpt 4915	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2766	. . . . 5 |- y e. _V
9	8, 8	opex 4263	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5155	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2200	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3634	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4896	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3851	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2208	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2868	. . . . . 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 2311	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2220	. 2 |- ran 2nd = _V
21		df-fo 5265	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 944	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1364   e. wcel 2167   {cab 2182   E.wrex 2476   _Vcvv 2763   {csn 3623   <.cop 3626   U.cuni 3840   ran crn 4665   Fn wfn 5254   -onto->wfo 5257   2ndc2nd 6206

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-fo 5265  df-2nd 6208

This theorem is referenced by:  2ndcof  6231  2ndexg  6235  df2nd2  6287  2ndconst  6289  suplocexprlemmu  7802  suplocexprlemdisj  7804  suplocexprlemloc  7805  suplocexprlemub  7807  upxp  14592  uptx  14594  cnmpt2nd  14609