Theorem fo1st 4082

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 3191	. . 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 2745	. . . . . 6 |- {x} e. V
3	2	dmex 3354	. . . . 5 |- dom { x} e. V
4	3	uniex 2865	. . . 4 |- U.dom { x} e. V
5		visset 1809	. . . . . 6 |- x e. V
6	5	biantrur 724	. . . . 5 |- (y = U.dom { x} <-> (x e. V /\ y = U.dom { x}))
7	6	opabbii 2666	. . . 4 |- {<.x, y>. | y = U.dom { x}} = {<.x, y>. | (x e. V /\ y = U.dom { x})}
8	4, 7	fnopab2 3611	. . 3 |- {<.x, y>. | y = U.dom { x}} Fn V
9		visset 1809	. . . . . . . . 9 |- y e. V
10	9	op1sta 3441	. . . . . . . 8 |- U.dom {<.y, y>.} = y
11	10	eqcomi 1476	. . . . . . 7 |- y = U.dom {<.y, y>.}
12		opex 2777	. . . . . . . 8 |- <.y, y>. e. V
13		sneq 2413	. . . . . . . . . . 11 |- (x = <.y, y>. -> {x} = {<.y, y>.})
14	13	dmeqd 3308	. . . . . . . . . 10 |- (x = <.y, y>. -> dom { x} = dom {<.y, y>.})
15	14	unieqd 2507	. . . . . . . . 9 |- (x = <.y, y>. -> U.dom { x} = U.dom {<.y, y>.})
16	15	eqeq2d 1483	. . . . . . . 8 |- (x = <.y, y>. -> (y = U.dom { x} <-> y = U.dom {<.y, y>.}))
17	12, 16	cla4ev 1865	. . . . . . 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 1124	. . . . . 6 |- y = y
20	18, 19	2th 717	. . . . 5 |- (E.x y = U.dom { x} <-> y = y)
21	20	abbii 1572	. . . 4 |- {y | E.x y = U.dom { x}} = {y | y = y}
22		rnopab 3347	. . . 4 |- ran {<.x, y>. | y = U.dom { x}} = {y | E.x y = U.dom { x}}
23		df-v 1808	. . . 4 |- V = {y | y = y}
24	21, 22, 23	3eqtr4 1502	. . 3 |- ran {<.x, y>. | y = U.dom { x}} = V
25	1, 8, 24	mpbir2an 729	. 2 |- {<.x, y>. | y = U.dom { x}}:V-onto->V
26		df-1st 4070	. . 3 |- 1st = {<.x, y>. | y = U.dom { x}}
27		foeq1 3660	. . 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 190	1 |- 1st:V-onto->V

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  Vcvv 1807  {csn 2405  <.cop 2407  U.cuni 2498  {copab 2661  dom cdm 3165  ran crn 3166   Fn wfn 3172  -onto->wfo 3175  1stc1st 4068

This theorem is referenced by:  1stcof 4092  df1st2 4117  ruclem10 7471  bcthlem3 7952  vafval 8175  smfval 8177  0vfval 8178  vsfval 8207  domval 10537  codval 10538  idval 10539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-fo 3191  df-1st 4070

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