Theorem 2ndval 4082

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 2750	. . . . 5 |- {A} e. V
2	1	rnex 3361	. . . 4 |- ran { A} e. V
3	2	uniex 2870	. . 3 |- U.ran { A} e. V
4		sneq 2417	. . . . . . 7 |- (x = A -> {x} = {A})
5	4	rneqd 3341	. . . . . 6 |- (x = A -> ran { x} = ran { A})
6	5	unieqd 2512	. . . . 5 |- (x = A -> U.ran { x} = U.ran { A})
7	6	fvopabg 3785	. . . 4 |- ((A e. V /\ U.ran { A} e. V) -> ({<.x, y>. | y = U.ran { x}}` A) = U.ran { A})
8		df-2nd 4080	. . . . 5 |- 2nd = {<.x, y>. | y = U.ran { x}}
9	8	fveq1i 3725	. . . 4 |- (2nd` A) = ({<.x, y>. | y = U.ran { x}}` A)
10	7, 9	syl5eq 1519	. . 3 |- ((A e. V /\ U.ran { A} e. V) -> (2nd` A) = U.ran { A})
11	3, 10	mpan2 696	. 2 |- (A e. V -> (2nd` A) = U.ran { A})
12		fvprc 3721	. . 3 |- (-. A e. V -> (2nd` A) = (/))
13		snprc 2443	. . . . . . . 8 |- (-. A e. V <-> {A} = (/))
14	13	biimp 151	. . . . . . 7 |- (-. A e. V -> {A} = (/))
15	14	rneqd 3341	. . . . . 6 |- (-. A e. V -> ran { A} = ran (/))
16		rn0 3355	. . . . . 6 |- ran (/) = (/)
17	15, 16	syl6eq 1523	. . . . 5 |- (-. A e. V -> ran { A} = (/))
18	17	unieqd 2512	. . . 4 |- (-. A e. V -> U.ran { A} = U.(/))
19		uni0 2525	. . . 4 |- U.(/) = (/)
20	18, 19	syl6eq 1523	. . 3 |- (-. A e. V -> U.ran { A} = (/))
21	12, 20	eqtr4d 1510	. 2 |- (-. A e. V -> (2nd` A) = U.ran { A})
22	11, 21	pm2.61i 126	1 |- (2nd` A) = U.ran { A}

Syntax hints:  -. wn 2   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  {csn 2409  U.cuni 2503  {copab 2666  ran crn 3171  ` cfv 3182  2ndc2nd 4078

This theorem is referenced by:  2nd0 4084  op2nd 4086  2nd2val 4096  elxp6 4102

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-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-rex 1650  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-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-2nd 4080

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