Theorem fo1st 6245

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 2775	. . . . . 6 |- x e. _V
2	1	snex 4230	. . . . 5 |- { x } e. _V
3	2	dmex 4946	. . . 4 |- dom { x } e. _V
4	3	uniex 4485	. . 3 |- U. dom { x } e. _V
5		df-1st 6228	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
6	4, 5	fnmpti 5406	. 2 |- 1st Fn _V
7	5	rnmpt 4927	. . 3 |- ran 1st = { y | E. x e. _V y = U. dom { x } }
8		vex 2775	. . . . 5 |- y e. _V
9	8, 8	opex 4274	. . . . . 6 |- <. y , y >. e. _V
10	8, 8	op1sta 5165	. . . . . . 7 |- U. dom { <. y , y >. } = y
11	10	eqcomi 2209	. . . . . 6 |- y = U. dom { <. y , y >. }
12		sneq 3644	. . . . . . . . . 10 |- ( x = <. y , y >. -> { x } = { <. y , y >. } )
13	12	dmeqd 4881	. . . . . . . . 9 |- ( x = <. y , y >. -> dom { x } = dom { <. y , y >. } )
14	13	unieqd 3861	. . . . . . . 8 |- ( x = <. y , y >. -> U. dom { x } = U. dom { <. y , y >. } )
15	14	eqeq2d 2217	. . . . . . 7 |- ( x = <. y , y >. -> ( y = U. dom { x } <-> y = U. dom { <. y , y >. } ) )
16	15	rspcev 2877	. . . . . 6 |- ( ( <. y , y >. e. _V /\ y = U. dom { <. y , y >. } ) -> E. x e. _V y = U. dom { x } )
17	9, 11, 16	mp2an 426	. . . . 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 2320	. . 3 |- _V = { y | E. x e. _V y = U. dom { x } }
20	7, 19	eqtr4i 2229	. 2 |- ran 1st = _V
21		df-fo 5278	. 2 |- ( 1st : _V -onto-> _V <-> ( 1st Fn _V /\ ran 1st = _V ) )
22	6, 20, 21	mpbir2an 945	1 |- 1st : _V -onto-> _V

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   {csn 3633   <.cop 3636   U.cuni 3850   dom cdm 4676   ran crn 4677   Fn wfn 5267   -onto->wfo 5270   1stc1st 6226

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 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-fun 5274  df-fn 5275  df-fo 5278  df-1st 6228

This theorem is referenced by:  1stcof  6251  1stexg  6255  df1st2  6307  1stconst  6309  algrflem  6317  algrflemg  6318  suplocexprlemell  7828  suplocexprlem2b  7829  suplocexprlemlub  7839  upxp  14777  uptx  14779  cnmpt1st  14793