Theorem fo2nd 6137

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 2733	. . . . . 6 |- x e. _V
2	1	snex 4171	. . . . 5 |- { x } e. _V
3	2	rnex 4878	. . . 4 |- ran { x } e. _V
4	3	uniex 4422	. . 3 |- U. ran { x } e. _V
5		df-2nd 6120	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5326	. 2 |- 2nd Fn _V
7	5	rnmpt 4859	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2733	. . . . 5 |- y e. _V
9	8, 8	opex 4214	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5095	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2174	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3594	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4840	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3807	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2182	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2834	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. ran { <. y , y >. } ) -> E. x e. _V y = U. ran { x } )
17	9, 11, 16	mp2an 424	. . . . 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 2285	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2194	. 2 |- ran 2nd = _V
21		df-fo 5204	. 2 |- ( 2nd : _V -onto-> _V <-> ( 2nd Fn _V /\ ran 2nd = _V ) )
22	6, 20, 21	mpbir2an 937	1 |- 2nd : _V -onto-> _V

Syntax hints:   = wceq 1348   e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730   {csn 3583   <.cop 3586   U.cuni 3796   ran crn 4612   Fn wfn 5193   -onto->wfo 5196   2ndc2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-fo 5204  df-2nd 6120

This theorem is referenced by:  2ndcof  6143  2ndexg  6147  df2nd2  6199  2ndconst  6201  suplocexprlemmu  7680  suplocexprlemdisj  7682  suplocexprlemloc  7683  suplocexprlemub  7685  upxp  13066  uptx  13068  cnmpt2nd  13083