Theorem 2ndval 3090

Description: The value of the function that extracts the second member of an ordered pair. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52. The notation is the same as Monk's.

Assertion

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

Proof of Theorem 2ndval

Step	Hyp	Ref	Expression
1		snex 1859	. . . . 5 |- {A} e. V
2		rnexg 2569	. . . . 5 |- ({A} e. V -> ran {A} e. V)
3	1, 2	ax-mp 6	. . . 4 |- ran {A} e. V
4	3	uniex 1947	. . 3 |- U.ran {A} e. V
5		sneq 1816	. . . . . . 7 |- (x = A -> {x} = {A})
6	5	rneqd 2557	. . . . . 6 |- (x = A -> ran {x} = ran {A})
7	6	unieqd 1929	. . . . 5 |- (x = A -> U.ran {x} = U.ran {A})
8	7	fvopabg 2872	. . . 4 |- ((A e. V /\ U.ran {A} e. V) -> ({<.x, y>. | y = U.ran {x}}` A) = U.ran {A})
9		df-2nd 3088	. . . . 5 |- 2nd = {<.x, y>. | y = U.ran {x}}
10	9	fveq1i 2833	. . . 4 |- (2nd` A) = ({<.x, y>. | y = U.ran {x}}` A)
11	8, 10	syl5eq 1136	. . 3 |- ((A e. V /\ U.ran {A} e. V) -> (2nd` A) = U.ran {A})
12	4, 11	mpan2 519	. 2 |- (A e. V -> (2nd` A) = U.ran {A})
13		fvprc 2829	. . 3 |- (-. A e. V -> (2nd` A) = (/))
14		snprc 1838	. . . . . . . 8 |- (-. A e. V <-> {A} = (/))
15	14	biimp 133	. . . . . . 7 |- (-. A e. V -> {A} = (/))
16	15	rneqd 2557	. . . . . 6 |- (-. A e. V -> ran {A} = ran (/))
17		rn0 2567	. . . . . 6 |- ran (/) = (/)
18	16, 17	syl6eq 1140	. . . . 5 |- (-. A e. V -> ran {A} = (/))
19	18	unieqd 1929	. . . 4 |- (-. A e. V -> U.ran {A} = U.(/))
20		uni0 1938	. . . 4 |- U.(/) = (/)
21	19, 20	syl6eq 1140	. . 3 |- (-. A e. V -> U.ran {A} = (/))
22	13, 21	eqtr4d 1131	. 2 |- (-. A e. V -> (2nd` A) = U.ran {A})
23	12, 22	pm2.61i 110	1 |- (2nd` A) = U.ran {A}

Syntax hints:  -. wn 1   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {csn 1808  U.cuni 1919  {copab 2055  ran crn 2411  ` cfv 2422  2ndc2nd 3086

This theorem is referenced by:  op2nd 3092  elxp6 3093

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-2nd 3088

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