Theorem fo2nd 4030

Description: The 2nd function maps the universe onto the universe.

Assertion

Ref	Expression
fo2nd	|- 2nd:V-onto->V

Proof of Theorem fo2nd

Step	Hyp	Ref	Expression
1		df-fo 3159	. . 3 |- ({<.x, y>. | y = U.ran { x}}:V-onto->V <-> ({<.x, y>. | y = U.ran { x}} Fn V /\ ran {<.x, y>. | y = U.ran { x}} = V))
2		snex 2718	. . . . . 6 |- {x} e. V
3		rnexg 3313	. . . . . 6 |- ({x} e. V -> ran { x} e. V)
4	2, 3	ax-mp 7	. . . . 5 |- ran { x} e. V
5	4	uniex 2834	. . . 4 |- U.ran { x} e. V
6		visset 1788	. . . . . 6 |- x e. V
7	6	biantrur 722	. . . . 5 |- (y = U.ran { x} <-> (x e. V /\ y = U.ran { x}))
8	7	opabbii 2639	. . . 4 |- {<.x, y>. | y = U.ran { x}} = {<.x, y>. | (x e. V /\ y = U.ran { x})}
9	5, 8	fnopab2 3558	. . 3 |- {<.x, y>. | y = U.ran { x}} Fn V
10		visset 1788	. . . . . . . . 9 |- y e. V
11	10, 10	op2nda 3401	. . . . . . . 8 |- U.ran {<.y, y>.} = y
12	11	eqcomi 1455	. . . . . . 7 |- y = U.ran {<.y, y>.}
13		opex 2750	. . . . . . . 8 |- <.y, y>. e. V
14		sneq 2388	. . . . . . . . . . 11 |- (x = <.y, y>. -> {x} = {<.y, y>.})
15	14	rneqd 3300	. . . . . . . . . 10 |- (x = <.y, y>. -> ran { x} = ran {<.y, y>.})
16	15	unieqd 2480	. . . . . . . . 9 |- (x = <.y, y>. -> U.ran { x} = U.ran {<.y, y>.})
17	16	eqeq2d 1462	. . . . . . . 8 |- (x = <.y, y>. -> (y = U.ran { x} <-> y = U.ran {<.y, y>.}))
18	13, 17	cla4ev 1842	. . . . . . 7 |- (y = U.ran {<.y, y>.} -> E.x y = U.ran { x})
19	12, 18	ax-mp 7	. . . . . 6 |- E.x y = U.ran { x}
20		equid 1113	. . . . . 6 |- y = y
21	19, 20	2th 715	. . . . 5 |- (E.x y = U.ran { x} <-> y = y)
22	21	abbii 1551	. . . 4 |- {y | E.x y = U.ran { x}} = {y | y = y}
23		rnopab 3309	. . . 4 |- ran {<.x, y>. | y = U.ran { x}} = {y | E.x y = U.ran { x}}
24		df-v 1787	. . . 4 |- V = {y | y = y}
25	22, 23, 24	3eqtr4 1481	. . 3 |- ran {<.x, y>. | y = U.ran { x}} = V
26	1, 9, 25	mpbir2an 727	. 2 |- {<.x, y>. | y = U.ran { x}}:V-onto->V
27		df-2nd 4018	. . 3 |- 2nd = {<.x, y>. | y = U.ran { x}}
28		foeq1 3607	. . 3 |- (2nd = {<.x, y>. | y = U.ran { x}} -> (2nd:V-onto->V <-> {<.x, y>. | y = U.ran { x}}:V-onto->V))
29	27, 28	ax-mp 7	. 2 |- (2nd:V-onto->V <-> {<.x, y>. | y = U.ran { x}}:V-onto->V)
30	26, 29	mpbir 190	1 |- 2nd:V-onto->V

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786  {csn 2380  <.cop 2382  U.cuni 2471  {copab 2634  ran crn 3134   Fn wfn 3140  -onto->wfo 3143  2ndc2nd 4016

This theorem is referenced by:  2ndconst 4035  df2nd2 4065  ruclem11 7414  smfval 8104  codval 8850  idval 8851  cmpval 8852

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-fun 3155  df-fn 3156  df-fo 3159  df-2nd 4018

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