Theorem fo1st 6381

Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fo1st	|- 1st : _V -onto-> _V

Proof of Theorem fo1st

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2824	. . . . . 6 |- x e. _V
2	1	snex 4317	. . . . 5 |- { x } e. _V
3	2	dmex 5044	. . . 4 |- dom { x } e. _V
4	3	uniex 4578	. . 3 |- U. dom { x } e. _V
5		df-1st 6364	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
6	4, 5	fnmpti 5507	. 2 |- 1st Fn _V
7	5	rnmpt 5025	. . 3 |- ran 1st = { y | E. x e. _V y = U. dom { x } }
8		vex 2824	. . . . 5 |- y e. _V
9	8, 8	opex 4364	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op1sta 5264	. . . . . . 7 |- U. dom { <. y , y >. } = y
11	10	eqcomi 2242	. . . . . 6 |- y = U. dom { <. y , y >. }
12		sneq 3716	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	dmeqd 4978	. . . . . . . . 9 |- ( x = <. y , y >. -> dom { x } = dom { <. y , y >. } )
14	13	unieqd 3941	. . . . . . . 8 |- ( x = <. y , y >. -> U. dom { x } = U. dom { <. y , y >. } )
15	14	eqeq2d 2250	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. dom { x } <-> y = U. dom { <. y , y >. } ) )
16	15	rspcev 2929	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. dom { <. y , y >. } ) -> E. x e. _V y = U. dom { x } )
17	9, 11, 16	mp2an 430	. . . . 5 |- E. x e. _V y = U. dom { x }
18	8, 17	2th 174	. . . 4 |- ( y e. _V <-> E. x e. _V y = U. dom { x } )
19	18	abbi2i 2353	. . 3 |- _V = { y | E. x e. _V y = U. dom { x } }
20	7, 19	eqtr4i 2262	. 2 |- ran 1st = _V
21		df-fo 5378	. 2 |- ( 1st : _V -onto-> _V <-> ( 1st Fn _V /\ ran 1st = _V ) )
22	6, 20, 21	mpbir2an 955	1 |- 1st : _V -onto-> _V

Syntax hints:   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821   {csn 3705   <.cop 3708   U.cuni 3930   dom cdm 4769   ran crn 4770   Fn wfn 5367   -onto->wfo 5370   1stc1st 6362

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-fo 5378  df-1st 6364

This theorem is referenced by:  1stcof  6387  1stexg  6391  df1st2  6445  1stconst  6447  algrflem  6455  algrflemg  6456  suplocexprlemell  8070  suplocexprlem2b  8071  suplocexprlemlub  8081  upxp  15296  uptx  15298  cnmpt1st  15312