Theorem fo2nd 6267

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 2779	. . . . . 6 |- x e. _V
2	1	snex 4245	. . . . 5 |- { x } e. _V
3	2	rnex 4965	. . . 4 |- ran { x } e. _V
4	3	uniex 4502	. . 3 |- U. ran { x } e. _V
5		df-2nd 6250	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
6	4, 5	fnmpti 5424	. 2 |- 2nd Fn _V
7	5	rnmpt 4945	. . 3 |- ran 2nd = { y | E. x e. _V y = U. ran { x } }
8		vex 2779	. . . . 5 |- y e. _V
9	8, 8	opex 4291	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op2nda 5186	. . . . . . 7 |- U. ran { <. y , y >. } = y
11	10	eqcomi 2211	. . . . . 6 |- y = U. ran { <. y , y >. }
12		sneq 3654	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	rneqd 4926	. . . . . . . . 9 |- ( x = <. y , y >. -> ran { x } = ran { <. y , y >. } )
14	13	unieqd 3875	. . . . . . . 8 |- ( x = <. y , y >. -> U. ran { x } = U. ran { <. y , y >. } )
15	14	eqeq2d 2219	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. ran { x } <-> y = U. ran { <. y , y >. } ) )
16	15	rspcev 2884	. . . . . 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 2322	. . 3 |- _V = { y | E. x e. _V y = U. ran { x } }
20	7, 19	eqtr4i 2231	. 2 |- ran 2nd = _V
21		df-fo 5296	. 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 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   {csn 3643   <.cop 3646   U.cuni 3864   ran crn 4694   Fn wfn 5285   -onto->wfo 5288   2ndc2nd 6248

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-fun 5292  df-fn 5293  df-fo 5296  df-2nd 6250

This theorem is referenced by:  2ndcof  6273  2ndexg  6277  df2nd2  6329  2ndconst  6331  suplocexprlemmu  7866  suplocexprlemdisj  7868  suplocexprlemloc  7869  suplocexprlemub  7871  upxp  14859  uptx  14861  cnmpt2nd  14876