Theorem fo1st 4169

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		df-fo 3251	. . 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))
2		snex 2802	. . . . . 6 |- {x} e. V
3	2	dmex 3420	. . . . 5 |- dom { x} e. V
4	3	uniex 2924	. . . 4 |- U.dom { x} e. V
5		visset 1851	. . . . . 6 |- x e. V
6	5	biantrur 728	. . . . 5 |- (y = U.dom { x} <-> (x e. V /\ y = U.dom { x}))
7	6	opabbii 2722	. . . 4 |- {<.x, y>. | y = U.dom { x}} = {<.x, y>. | (x e. V /\ y = U.dom { x})}
8	4, 7	fnopab2 3693	. . 3 |- {<.x, y>. | y = U.dom { x}} Fn V
9		visset 1851	. . . . . . . . 9 |- y e. V
10	9	op1sta 3550	. . . . . . . 8 |- U.dom {<.y, y>.} = y
11	10	eqcomi 1516	. . . . . . 7 |- y = U.dom {<.y, y>.}
12		opex 2835	. . . . . . . 8 |- <.y, y>. e. V
13		sneq 2462	. . . . . . . . . . 11 |- (x = <.y, y>. -> {x} = {<.y, y>.})
14	13	dmeqd 3377	. . . . . . . . . 10 |- (x = <.y, y>. -> dom { x} = dom {<.y, y>.})
15	14	unieqd 2560	. . . . . . . . 9 |- (x = <.y, y>. -> U.dom { x} = U.dom {<.y, y>.})
16	15	eqeq2d 1523	. . . . . . . 8 |- (x = <.y, y>. -> (y = U.dom { x} <-> y = U.dom {<.y, y>.}))
17	12, 16	cla4ev 1907	. . . . . . 7 |- (y = U.dom {<.y, y>.} -> E.x y = U.dom { x})
18	11, 17	ax-mp 7	. . . . . 6 |- E.x y = U.dom { x}
19		equid 1158	. . . . . 6 |- y = y
20	18, 19	2th 721	. . . . 5 |- (E.x y = U.dom { x} <-> y = y)
21	20	abbii 1612	. . . 4 |- {y | E.x y = U.dom { x}} = {y | y = y}
22		rnopab 3413	. . . 4 |- ran {<.x, y>. | y = U.dom { x}} = {y | E.x y = U.dom { x}}
23		df-v 1850	. . . 4 |- V = {y | y = y}
24	21, 22, 23	3eqtr4i 1542	. . 3 |- ran {<.x, y>. | y = U.dom { x}} = V
25	1, 8, 24	mpbir2an 733	. 2 |- {<.x, y>. | y = U.dom { x}}:V-onto->V
26		df-1st 4157	. . 3 |- 1st = {<.x, y>. | y = U.dom { x}}
27		foeq1 3743	. . 3 |- (1st = {<.x, y>. | y = U.dom { x}} -> (1st:V-onto->V <-> {<.x, y>. | y = U.dom { x}}:V-onto->V))
28	26, 27	ax-mp 7	. 2 |- (1st:V-onto->V <-> {<.x, y>. | y = U.dom { x}}:V-onto->V)
29	25, 28	mpbir 188	1 |- 1st:V-onto->V

Syntax hints:   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  E.wex 1012  {cab 1499  Vcvv 1849  {csn 2454  <.cop 2456  U.cuni 2551  {copab 2717  dom cdm 3225  ran crn 3226   Fn wfn 3232  -onto->wfo 3235  1stc1st 4155

This theorem is referenced by:  1stcof 4177  1stconst 4206  ruclem10 7644  bcthlem3 8121  vafval 8341  smfval 8343  0vfval 8344  vsfval 8373  domval 10790  codval 10791  idval 10792

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-fun 3247  df-fn 3248  df-fo 3251  df-1st 4157

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