Theorem fo1st 3094

Description: The 1st function maps the universe onto the universe.

Assertion

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

Proof of Theorem fo1st

Step	Hyp	Ref	Expression
1		snex 1859	. . . . . . 7 |- {x} e. V
2		dmexg 2551	. . . . . . 7 |- ({x} e. V -> dom {x} e. V)
3	1, 2	ax-mp 6	. . . . . 6 |- dom {x} e. V
4	3	uniex 1947	. . . . 5 |- U.dom {x} e. V
5		visset 1350	. . . . . . 7 |- x e. V
6	5	biantrur 544	. . . . . 6 |- (y = U.dom {x} <-> (x e. V /\ y = U.dom {x}))
7	6	biopabi 2103	. . . . 5 |- {<.x, y>. | y = U.dom {x}} = {<.x, y>. | (x e. V /\ y = U.dom {x})}
8	4, 7	fnopab2 2747	. . . 4 |- {<.x, y>. | y = U.dom {x}} Fn V
9		visset 1350	. . . . . . . . . 10 |- y e. V
10	9	op1sta 2635	. . . . . . . . 9 |- U.dom {<.y, y>.} = y
11	10	cleqcomi 1105	. . . . . . . 8 |- y = U.dom {<.y, y>.}
12		opex 1893	. . . . . . . . 9 |- <.y, y>. e. V
13		sneq 1816	. . . . . . . . . . . 12 |- (x = <.y, y>. -> {x} = {<.y, y>.})
14	13	dmeqd 2533	. . . . . . . . . . 11 |- (x = <.y, y>. -> dom {x} = dom {<.y, y>.})
15	14	unieqd 1929	. . . . . . . . . 10 |- (x = <.y, y>. -> U.dom {x} = U.dom {<.y, y>.})
16	15	cleq2d 1112	. . . . . . . . 9 |- (x = <.y, y>. -> (y = U.dom {x} <-> y = U.dom {<.y, y>.}))
17	12, 16	cla4ev 1401	. . . . . . . 8 |- (y = U.dom {<.y, y>.} -> E.x y = U.dom {x})
18	11, 17	ax-mp 6	. . . . . . 7 |- E.x y = U.dom {x}
19		eqid 810	. . . . . . 7 |- y = y
20	18, 19	2th 540	. . . . . 6 |- (E.x y = U.dom {x} <-> y = y)
21	20	biabi 1181	. . . . 5 |- {y | E.x y = U.dom {x}} = {y | y = y}
22		rnopab 2566	. . . . 5 |- ran {<.x, y>. | y = U.dom {x}} = {y | E.x y = U.dom {x}}
23		df-v 1349	. . . . 5 |- V = {y | y = y}
24	21, 22, 23	3eqtr4 1126	. . . 4 |- ran {<.x, y>. | y = U.dom {x}} = V
25	8, 24	pm3.2i 234	. . 3 |- ({<.x, y>. | y = U.dom {x}} Fn V /\ ran {<.x, y>. | y = U.dom {x}} = V)
26		df-fo 2436	. . 3 |- ({<.x, y>. | y = U.dom {x}}:V-onto->V <-> ({<.x, y>. | y = U.dom {x}} Fn V /\ ran {<.x, y>. | y = U.dom {x}} = V))
27	25, 26	mpbir 165	. 2 |- {<.x, y>. | y = U.dom {x}}:V-onto->V
28		df-1st 3087	. . 3 |- 1st = {<.x, y>. | y = U.dom {x}}
29		foeq1 2784	. . 3 |- (1st = {<.x, y>. | y = U.dom {x}} -> (1st:V-onto->V <-> {<.x, y>. | y = U.dom {x}}:V-onto->V))
30	28, 29	ax-mp 6	. 2 |- (1st:V-onto->V <-> {<.x, y>. | y = U.dom {x}}:V-onto->V)
31	27, 30	mpbir 165	1 |- 1st:V-onto->V

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  U.cuni 1919  {copab 2055  dom cdm 2410  ran crn 2411   Fn wfn 2417  -onto->wfo 2420  1stc1st 3085

This theorem is referenced by:  df1st2 3098  ruclem10 4894

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-fo 2436  df-1st 3087

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