Theorem 2ndval 5023

Description: The value of the function that extracts the second member of an ordered pair.

Assertion

Ref	Expression
2ndval	|- (2nd` A) = U.ran { A}

Proof of Theorem 2ndval

Step	Hyp	Ref	Expression
1		snex 3492	. . . . 5 |- {A} e. _V
2	1	rnex 4209	. . . 4 |- ran { A} e. _V
3	2	uniex 3794	. . 3 |- U.ran { A} e. _V
4		sneq 3054	. . . . . . 7 |- (x = A -> {x} = {A})
5	4	rneqd 4188	. . . . . 6 |- (x = A -> ran { x} = ran { A})
6	5	unieqd 3188	. . . . 5 |- (x = A -> U.ran { x} = U.ran { A})
7	6	fvopabg 4748	. . . 4 |- ((A e. _V /\ U.ran { A} e. _V) -> ({<.x, y>. | y = U.ran { x}}` A) = U.ran { A})
8		df-2nd 5021	. . . . 5 |- 2nd = {<.x, y>. | y = U.ran { x}}
9	8	fveq1i 4682	. . . 4 |- (2nd` A) = ({<.x, y>. | y = U.ran { x}}` A)
10	7, 9	syl5eq 1940	. . 3 |- ((A e. _V /\ U.ran { A} e. _V) -> (2nd` A) = U.ran { A})
11	3, 10	mpan2 760	. 2 |- (A e. _V -> (2nd` A) = U.ran { A})
12		fvprc 4678	. . 3 |- (-. A e. _V -> (2nd` A) = (/))
13		snprc 3092	. . . . . . . 8 |- (-. A e. _V <-> {A} = (/))
14	13	biimpi 168	. . . . . . 7 |- (-. A e. _V -> {A} = (/))
15	14	rneqd 4188	. . . . . 6 |- (-. A e. _V -> ran { A} = ran (/))
16		rn0 4203	. . . . . 6 |- ran (/) = (/)
17	15, 16	syl6eq 1944	. . . . 5 |- (-. A e. _V -> ran { A} = (/))
18	17	unieqd 3188	. . . 4 |- (-. A e. _V -> U.ran { A} = U.(/))
19		uni0 3205	. . . 4 |- U.(/) = (/)
20	18, 19	syl6eq 1944	. . 3 |- (-. A e. _V -> U.ran { A} = (/))
21	12, 20	eqtr4d 1928	. 2 |- (-. A e. _V -> (2nd` A) = U.ran { A})
22	11, 21	pm2.61i 140	1 |- (2nd` A) = U.ran { A}

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  {csn 3044  U.cuni 3177  {copab 3395  ran crn 3987  ` cfv 3998  2ndc2nd 5019

This theorem is referenced by:  2nd0 5025  op2nd 5027  2nd2val 5039  elxp6 5041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-2nd 5021

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