Theorem unirep 15664

Description: Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice.

Hypotheses

Ref	Expression
unirep.1	|- (y = D -> (ph <-> ps))
unirep.2	|- (y = D -> B = C)
unirep.3	|- (y = z -> (ph <-> ch))
unirep.4	|- (y = z -> B = F)
unirep.5	|- B e. _V

Assertion

Ref	Expression
unirep	|- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> U.{x | E.y e. A (ph /\ x = B)} = C)

Distinct variable groups:   x,A,y,z   x,B,z   x,C,y   x,D,y   x,F,y   ph,x,z   ps,x,y   ch,x,y

Proof of Theorem unirep

Step	Hyp	Ref	Expression
1		unirep.1	. . . . . 6 |- (y = D -> (ph <-> ps))
2		unirep.2	. . . . . . 7 |- (y = D -> B = C)
3	2	eqeq2d 1895	. . . . . 6 |- (y = D -> (C = B <-> C = C))
4	1, 3	anbi12d 690	. . . . 5 |- (y = D -> ((ph /\ C = B) <-> (ps /\ C = C)))
5	4	rcla4ev 2381	. . . 4 |- ((D e. A /\ (ps /\ C = C)) -> E.y e. A (ph /\ C = B))
6		eqidd 1885	. . . . 5 |- (ps -> C = C)
7	6	ancli 320	. . . 4 |- (ps -> (ps /\ C = C))
8	5, 7	sylan2 500	. . 3 |- ((D e. A /\ ps) -> E.y e. A (ph /\ C = B))
9	8	adantl 424	. 2 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> E.y e. A (ph /\ C = B))
10		ax-17 1317	. . . . . . 7 |- (w e. C -> A.y w e. C)
11	10	a1i 8	. . . . . 6 |- (D e. A -> (w e. C -> A.y w e. C))
12	11, 2	csbiegf 2575	. . . . 5 |- (D e. A -> [_D / y]_B = C)
13		unirep.5	. . . . . . 7 |- B e. _V
14	13	ax-gen 1305	. . . . . 6 |- A.y B e. _V
15		csbexg 2548	. . . . . 6 |- ((D e. A /\ A.y B e. _V) -> [_D / y]_B e. _V)
16	14, 15	mpan2 760	. . . . 5 |- (D e. A -> [_D / y]_B e. _V)
17	12, 16	eqeltrrd 1972	. . . 4 |- (D e. A -> C e. _V)
18	17	ad2antrl 442	. . 3 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> C e. _V)
19		eqeq1 1890	. . . . . . . 8 |- (x = w -> (x = B <-> w = B))
20	19	anbi2d 678	. . . . . . 7 |- (x = w -> ((ph /\ x = B) <-> (ph /\ w = B)))
21	20	rexbidv 2124	. . . . . 6 |- (x = w -> (E.y e. A (ph /\ x = B) <-> E.y e. A (ph /\ w = B)))
22		unirep.3	. . . . . . . 8 |- (y = z -> (ph <-> ch))
23		unirep.4	. . . . . . . . 9 |- (y = z -> B = F)
24	23	eqeq2d 1895	. . . . . . . 8 |- (y = z -> (w = B <-> w = F))
25	22, 24	anbi12d 690	. . . . . . 7 |- (y = z -> ((ph /\ w = B) <-> (ch /\ w = F)))
26	25	cbvrexv 2281	. . . . . 6 |- (E.y e. A (ph /\ w = B) <-> E.z e. A (ch /\ w = F))
27	21, 26	syl6bb 595	. . . . 5 |- (x = w -> (E.y e. A (ph /\ x = B) <-> E.z e. A (ch /\ w = F)))
28	27	eu4 1806	. . . 4 |- (E!xE.y e. A (ph /\ x = B) <-> (E.xE.y e. A (ph /\ x = B) /\ A.xA.w((E.y e. A (ph /\ x = B) /\ E.z e. A (ch /\ w = F)) -> x = w)))
29		eqeq1 1890	. . . . . . . . . . 11 |- (x = C -> (x = B <-> C = B))
30	29	anbi2d 678	. . . . . . . . . 10 |- (x = C -> ((ph /\ x = B) <-> (ph /\ C = B)))
31	30	rexbidv 2124	. . . . . . . . 9 |- (x = C -> (E.y e. A (ph /\ x = B) <-> E.y e. A (ph /\ C = B)))
32	31	cla4egv 2365	. . . . . . . 8 |- (C e. _V -> (E.y e. A (ph /\ C = B) -> E.xE.y e. A (ph /\ x = B)))
33	17, 32	syl 12	. . . . . . 7 |- (D e. A -> (E.y e. A (ph /\ C = B) -> E.xE.y e. A (ph /\ x = B)))
34	33	adantr 425	. . . . . 6 |- ((D e. A /\ ps) -> (E.y e. A (ph /\ C = B) -> E.xE.y e. A (ph /\ x = B)))
35	8, 34	mpd 29	. . . . 5 |- ((D e. A /\ ps) -> E.xE.y e. A (ph /\ x = B))
36	35	adantl 424	. . . 4 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> E.xE.y e. A (ph /\ x = B))
37		r19.29 2227	. . . . . . . 8 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ E.y e. A (ph /\ x = B)) -> E.y e. A (A.z e. A ((ph /\ ch) -> B = F) /\ (ph /\ x = B)))
38		eqeq12 1896	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x = B /\ w = F) -> (x = w <-> B = F))
39		pm3.35 386	. . . . . . . . . . . . . . . . . . . . . 22 |- (((ph /\ ch) /\ ((ph /\ ch) -> B = F)) -> B = F)
40	38, 39	syl5cbir 228	. . . . . . . . . . . . . . . . . . . . 21 |- (((ph /\ ch) /\ ((ph /\ ch) -> B = F)) -> ((x = B /\ w = F) -> x = w))
41	40	ancoms 484	. . . . . . . . . . . . . . . . . . . 20 |- ((((ph /\ ch) -> B = F) /\ (ph /\ ch)) -> ((x = B /\ w = F) -> x = w))
42	41	expimpd 404	. . . . . . . . . . . . . . . . . . 19 |- (((ph /\ ch) -> B = F) -> (((ph /\ ch) /\ (x = B /\ w = F)) -> x = w))
43		an4 564	. . . . . . . . . . . . . . . . . . 19 |- (((ph /\ x = B) /\ (ch /\ w = F)) <-> ((ph /\ ch) /\ (x = B /\ w = F)))
44	42, 43	syl5ib 223	. . . . . . . . . . . . . . . . . 18 |- (((ph /\ ch) -> B = F) -> (((ph /\ x = B) /\ (ch /\ w = F)) -> x = w))
45	44	ancomsd 485	. . . . . . . . . . . . . . . . 17 |- (((ph /\ ch) -> B = F) -> (((ch /\ w = F) /\ (ph /\ x = B)) -> x = w))
46	45	expdimp 406	. . . . . . . . . . . . . . . 16 |- ((((ph /\ ch) -> B = F) /\ (ch /\ w = F)) -> ((ph /\ x = B) -> x = w))
47	46	a1i 8	. . . . . . . . . . . . . . 15 |- (z e. A -> ((((ph /\ ch) -> B = F) /\ (ch /\ w = F)) -> ((ph /\ x = B) -> x = w)))
48	47	r19.23aiv 2211	. . . . . . . . . . . . . 14 |- (E.z e. A (((ph /\ ch) -> B = F) /\ (ch /\ w = F)) -> ((ph /\ x = B) -> x = w))
49	48	imp 377	. . . . . . . . . . . . 13 |- ((E.z e. A (((ph /\ ch) -> B = F) /\ (ch /\ w = F)) /\ (ph /\ x = B)) -> x = w)
50		r19.29 2227	. . . . . . . . . . . . 13 |- ((A.z e. A ((ph /\ ch) -> B = F) /\ E.z e. A (ch /\ w = F)) -> E.z e. A (((ph /\ ch) -> B = F) /\ (ch /\ w = F)))
51	49, 50	sylan 497	. . . . . . . . . . . 12 |- (((A.z e. A ((ph /\ ch) -> B = F) /\ E.z e. A (ch /\ w = F)) /\ (ph /\ x = B)) -> x = w)
52	51	an1rs 547	. . . . . . . . . . 11 |- (((A.z e. A ((ph /\ ch) -> B = F) /\ (ph /\ x = B)) /\ E.z e. A (ch /\ w = F)) -> x = w)
53	52	ex 402	. . . . . . . . . 10 |- ((A.z e. A ((ph /\ ch) -> B = F) /\ (ph /\ x = B)) -> (E.z e. A (ch /\ w = F) -> x = w))
54	53	a1i 8	. . . . . . . . 9 |- (y e. A -> ((A.z e. A ((ph /\ ch) -> B = F) /\ (ph /\ x = B)) -> (E.z e. A (ch /\ w = F) -> x = w)))
55	54	r19.23aiv 2211	. . . . . . . 8 |- (E.y e. A (A.z e. A ((ph /\ ch) -> B = F) /\ (ph /\ x = B)) -> (E.z e. A (ch /\ w = F) -> x = w))
56	37, 55	syl 12	. . . . . . 7 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ E.y e. A (ph /\ x = B)) -> (E.z e. A (ch /\ w = F) -> x = w))
57	56	expimpd 404	. . . . . 6 |- (A.y e. A A.z e. A ((ph /\ ch) -> B = F) -> ((E.y e. A (ph /\ x = B) /\ E.z e. A (ch /\ w = F)) -> x = w))
58	57	adantr 425	. . . . 5 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> ((E.y e. A (ph /\ x = B) /\ E.z e. A (ch /\ w = F)) -> x = w))
59	58	19.21aivv 1665	. . . 4 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> A.xA.w((E.y e. A (ph /\ x = B) /\ E.z e. A (ch /\ w = F)) -> x = w))
60	28, 36, 59	sylanbrc 527	. . 3 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> E!xE.y e. A (ph /\ x = B))
61	31	euuni2 15663	. . 3 |- ((C e. _V /\ E!xE.y e. A (ph /\ x = B)) -> (E.y e. A (ph /\ C = B) <-> U.{x | E.y e. A (ph /\ x = B)} = C))
62	18, 60, 61	syl11anc 524	. 2 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> (E.y e. A (ph /\ C = B) <-> U.{x | E.y e. A (ph /\ x = B)} = C))
63	9, 62	mpbid 212	1 |- ((A.y e. A A.z e. A ((ph /\ ch) -> B = F) /\ (D e. A /\ ps)) -> U.{x | E.y e. A (ph /\ x = B)} = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292  [_csb 2540  U.cuni 3177

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-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  df-reu 2111  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-uni 3178

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