Theorem isseg2 15305

Description: The segment <.Q, R>..

Hypothesis

Ref	Expression
isseg2.1	|- P = U.L

Assertion

Ref	Expression
isseg2	|- ((L e. A /\ B e. C /\ (Q e. P /\ R e. P)) -> (Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})

Distinct variable groups:   B,p   P,p   Q,p   R,p

Proof of Theorem isseg2

Step	Hyp	Ref	Expression
1		isseg2.1	. . . 4 |- P = U.L
2	1	isseg1 15304	. . 3 |- ((L e. A /\ B e. C) -> (seg` <.L, B>.) = {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})})
3		opreq 4888	. . . . . 6 |- ((seg` <.L, B>.) = {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})} -> (Q(seg` <.L, B>.)R) = (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R))
4	3	eqeq1d 1892	. . . . 5 |- ((seg` <.L, B>.) = {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})} -> ((Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} <-> (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)}))
5	4	imbi2d 674	. . . 4 |- ((seg` <.L, B>.) = {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})} -> (((Q e. P /\ R e. P) -> (Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)}) <-> ((Q e. P /\ R e. P) -> (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})))
6		opeq1 3158	. . . . . . . . . . 11 |- (a = Q -> <.a, p>. = <.Q, p>.)
7	6	opeq1d 3164	. . . . . . . . . 10 |- (a = Q -> <.<.a, p>., b>. = <.<.Q, p>., b>.)
8	7	eleq1d 1963	. . . . . . . . 9 |- (a = Q -> (<.<.a, p>., b>. e. B <-> <.<.Q, p>., b>. e. B))
9		eqeq2 1893	. . . . . . . . 9 |- (a = Q -> (p = a <-> p = Q))
10		biidd 188	. . . . . . . . 9 |- (a = Q -> (p = b <-> p = b))
11	8, 9, 10	3orbi123d 1167	. . . . . . . 8 |- (a = Q -> ((<.<.a, p>., b>. e. B \/ p = a \/ p = b) <-> (<.<.Q, p>., b>. e. B \/ p = Q \/ p = b)))
12	11	rabbidv 2287	. . . . . . 7 |- (a = Q -> {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)} = {p e. P | (<.<.Q, p>., b>. e. B \/ p = Q \/ p = b)})
13		opeq2 3159	. . . . . . . . . 10 |- (b = R -> <.<.Q, p>., b>. = <.<.Q, p>., R>.)
14	13	eleq1d 1963	. . . . . . . . 9 |- (b = R -> (<.<.Q, p>., b>. e. B <-> <.<.Q, p>., R>. e. B))
15		biidd 188	. . . . . . . . 9 |- (b = R -> (p = Q <-> p = Q))
16		eqeq2 1893	. . . . . . . . 9 |- (b = R -> (p = b <-> p = R))
17	14, 15, 16	3orbi123d 1167	. . . . . . . 8 |- (b = R -> ((<.<.Q, p>., b>. e. B \/ p = Q \/ p = b) <-> (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)))
18	17	rabbidv 2287	. . . . . . 7 |- (b = R -> {p e. P | (<.<.Q, p>., b>. e. B \/ p = Q \/ p = b)} = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})
19		df-3an 860	. . . . . . . 8 |- ((a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)}) <-> ((a e. P /\ b e. P) /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)}))
20	19	oprabbii 4923	. . . . . . 7 |- {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})} = {<.<.a, b>., c>. | ((a e. P /\ b e. P) /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}
21	12, 18, 20	oprabval2g 4956	. . . . . 6 |- ((Q e. P /\ R e. P /\ {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} e. _V) -> (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})
22	21	3expia 1069	. . . . 5 |- ((Q e. P /\ R e. P) -> ({p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} e. _V -> (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)}))
23		uniexg 3795	. . . . . . . 8 |- (L e. A -> U.L e. _V)
24	23, 1	syl5eqel 1975	. . . . . . 7 |- (L e. A -> P e. _V)
25		rabexg 3460	. . . . . . 7 |- (P e. _V -> {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} e. _V)
26	24, 25	syl 12	. . . . . 6 |- (L e. A -> {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} e. _V)
27	26	adantr 425	. . . . 5 |- ((L e. A /\ B e. C) -> {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)} e. _V)
28	22, 27	syl5com 63	. . . 4 |- ((L e. A /\ B e. C) -> ((Q e. P /\ R e. P) -> (Q{<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})}R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)}))
29	5, 28	syl5bir 227	. . 3 |- ((seg` <.L, B>.) = {<.<.a, b>., c>. | (a e. P /\ b e. P /\ c = {p e. P | (<.<.a, p>., b>. e. B \/ p = a \/ p = b)})} -> ((L e. A /\ B e. C) -> ((Q e. P /\ R e. P) -> (Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})))
30	2, 29	mpcom 60	. 2 |- ((L e. A /\ B e. C) -> ((Q e. P /\ R e. P) -> (Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)}))
31	30	3impia 1064	1 |- ((L e. A /\ B e. C /\ (Q e. P /\ R e. P)) -> (Q(seg` <.L, B>.)R) = {p e. P | (<.<.Q, p>., R>. e. B \/ p = Q \/ p = R)})

Syntax hints:   -> wi 3   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292  <.cop 3046  U.cuni 3177  ` cfv 3998  (class class class)co 4884  {copab2 4885  segcseg 15302

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-rep 3428  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-3or 859  df-3an 860  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-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-seg 15303

Copyright terms: Public domain