Theorem fo2nd 4092

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 3196	. . 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 2750	. . . . . 6 |- {x} e. V
3	2	rnex 3361	. . . . 5 |- ran { x} e. V
4	3	uniex 2870	. . . 4 |- U.ran { x} e. V
5		visset 1813	. . . . . 6 |- x e. V
6	5	biantrur 725	. . . . 5 |- (y = U.ran { x} <-> (x e. V /\ y = U.ran { x}))
7	6	opabbii 2671	. . . 4 |- {<.x, y>. | y = U.ran { x}} = {<.x, y>. | (x e. V /\ y = U.ran { x})}
8	4, 7	fnopab2 3618	. . 3 |- {<.x, y>. | y = U.ran { x}} Fn V
9		visset 1813	. . . . . . . . 9 |- y e. V
10	9, 9	op2nda 3452	. . . . . . . 8 |- U.ran {<.y, y>.} = y
11	10	eqcomi 1479	. . . . . . 7 |- y = U.ran {<.y, y>.}
12		opex 2782	. . . . . . . 8 |- <.y, y>. e. V
13		sneq 2417	. . . . . . . . . . 11 |- (x = <.y, y>. -> {x} = {<.y, y>.})
14	13	rneqd 3341	. . . . . . . . . 10 |- (x = <.y, y>. -> ran { x} = ran {<.y, y>.})
15	14	unieqd 2512	. . . . . . . . 9 |- (x = <.y, y>. -> U.ran { x} = U.ran {<.y, y>.})
16	15	eqeq2d 1486	. . . . . . . 8 |- (x = <.y, y>. -> (y = U.ran { x} <-> y = U.ran {<.y, y>.}))
17	12, 16	cla4ev 1869	. . . . . . 7 |- (y = U.ran {<.y, y>.} -> E.x y = U.ran { x})
18	11, 17	ax-mp 7	. . . . . 6 |- E.x y = U.ran { x}
19		equid 1126	. . . . . 6 |- y = y
20	18, 19	2th 718	. . . . 5 |- (E.x y = U.ran { x} <-> y = y)
21	20	abbii 1575	. . . 4 |- {y | E.x y = U.ran { x}} = {y | y = y}
22		rnopab 3353	. . . 4 |- ran {<.x, y>. | y = U.ran { x}} = {y | E.x y = U.ran { x}}
23		df-v 1812	. . . 4 |- V = {y | y = y}
24	21, 22, 23	3eqtr4 1505	. . 3 |- ran {<.x, y>. | y = U.ran { x}} = V
25	1, 8, 24	mpbir2an 730	. 2 |- {<.x, y>. | y = U.ran { x}}:V-onto->V
26		df-2nd 4080	. . 3 |- 2nd = {<.x, y>. | y = U.ran { x}}
27		foeq1 3668	. . 3 |- (2nd = {<.x, y>. | y = U.ran { x}} -> (2nd:V-onto->V <-> {<.x, y>. | y = U.ran { x}}:V-onto->V))
28	26, 27	ax-mp 7	. 2 |- (2nd:V-onto->V <-> {<.x, y>. | y = U.ran { x}}:V-onto->V)
29	25, 28	mpbir 190	1 |- 2nd:V-onto->V

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811  {csn 2409  <.cop 2411  U.cuni 2503  {copab 2666  ran crn 3171   Fn wfn 3177  -onto->wfo 3180  2ndc2nd 4078

This theorem is referenced by:  2ndconst 4097  df2nd2 4127  ruclem11 7520  smfval 8224  codval 10656  idval 10657  cmpval 10658

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-fo 3196  df-2nd 4080

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