Theorem cncnplem4 9054

Description: Lemma for cncnp2 9056.

Hypothesis

Ref	Expression
cncnplem4.1	|- X = U.J

Assertion

Ref	Expression
cncnplem4	|- (J e. Top -> ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> (`'F"y) e. J))

Distinct variable groups:   x,J,z   x,F,z   x,X,z   x,Y,z   x,y,z

Proof of Theorem cncnplem4

Step	Hyp	Ref	Expression
1		fdm 4567	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> dom F = X)
2		cncnplem4.1	. . . . . . . . . . . . . . 15 |- X = U.J
3	1, 2	syl6eq 1944	. . . . . . . . . . . . . 14 |- (F:X-->Y -> dom F = U.J)
4	3	sseq2d 2645	. . . . . . . . . . . . 13 |- (F:X-->Y -> (z C_ dom F <-> z C_ U.J))
5		elssuni 3206	. . . . . . . . . . . . 13 |- (z e. J -> z C_ U.J)
6	4, 5	syl5bir 227	. . . . . . . . . . . 12 |- (F:X-->Y -> (z e. J -> z C_ dom F))
7		funimass3 4779	. . . . . . . . . . . . . 14 |- ((Fun F /\ z C_ dom F) -> ((F"z) C_ y <-> z C_ (`'F"y)))
8		ffun 4565	. . . . . . . . . . . . . 14 |- (F:X-->Y -> Fun F)
9	7, 8	sylan 497	. . . . . . . . . . . . 13 |- ((F:X-->Y /\ z C_ dom F) -> ((F"z) C_ y <-> z C_ (`'F"y)))
10	9	ex 402	. . . . . . . . . . . 12 |- (F:X-->Y -> (z C_ dom F -> ((F"z) C_ y <-> z C_ (`'F"y))))
11	6, 10	syld 30	. . . . . . . . . . 11 |- (F:X-->Y -> (z e. J -> ((F"z) C_ y <-> z C_ (`'F"y))))
12	11	imp 377	. . . . . . . . . 10 |- ((F:X-->Y /\ z e. J) -> ((F"z) C_ y <-> z C_ (`'F"y)))
13	12	anbi2d 678	. . . . . . . . 9 |- ((F:X-->Y /\ z e. J) -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ z C_ (`'F"y))))
14	13	rabbidva 2286	. . . . . . . 8 |- (F:X-->Y -> {z e. J | (x e. z /\ (F"z) C_ y)} = {z e. J | (x e. z /\ z C_ (`'F"y))})
15	14	unieqd 3188	. . . . . . 7 |- (F:X-->Y -> U.{z e. J | (x e. z /\ (F"z) C_ y)} = U.{z e. J | (x e. z /\ z C_ (`'F"y))})
16	15	adantr 425	. . . . . 6 |- ((F:X-->Y /\ x e. (`'F"y)) -> U.{z e. J | (x e. z /\ (F"z) C_ y)} = U.{z e. J | (x e. z /\ z C_ (`'F"y))})
17	16	iuneq2dv 3279	. . . . 5 |- (F:X-->Y -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} = U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))})
18	17	adantr 425	. . . 4 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} = U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))})
19		iunss 3291	. . . . . . 7 |- (U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y) <-> A.x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y))
20		cncnplem1 9051	. . . . . . . 8 |- U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y)
21	20	a1i 8	. . . . . . 7 |- (x e. (`'F"y) -> U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y))
22	19, 21	mprgbir 2163	. . . . . 6 |- U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y)
23	22	a1i 8	. . . . 5 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))} C_ (`'F"y))
24		cnvimass 4286	. . . . . . . 8 |- (`'F"y) C_ dom F
25	1	sseq2d 2645	. . . . . . . 8 |- (F:X-->Y -> ((`'F"y) C_ dom F <-> (`'F"y) C_ X))
26	24, 25	mpbii 210	. . . . . . 7 |- (F:X-->Y -> (`'F"y) C_ X)
27	24	sseli 2617	. . . . . . . . . 10 |- (x e. (`'F"y) -> x e. dom F)
28		fvimacnv 4778	. . . . . . . . . . . . . . 15 |- ((Fun F /\ x e. dom F) -> ((F` x) e. y <-> x e. (`'F"y)))
29	28, 8	sylan 497	. . . . . . . . . . . . . 14 |- ((F:X-->Y /\ x e. dom F) -> ((F` x) e. y <-> x e. (`'F"y)))
30	29	biimprd 171	. . . . . . . . . . . . 13 |- ((F:X-->Y /\ x e. dom F) -> (x e. (`'F"y) -> (F` x) e. y))
31	9	anbi2d 678	. . . . . . . . . . . . . . . . . . . . 21 |- ((F:X-->Y /\ z C_ dom F) -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ z C_ (`'F"y))))
32		pm4.24 479	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. z <-> (x e. z /\ x e. z))
33	32	anbi1i 539	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. z /\ z C_ (`'F"y)) <-> ((x e. z /\ x e. z) /\ z C_ (`'F"y)))
34		anass 487	. . . . . . . . . . . . . . . . . . . . . 22 |- (((x e. z /\ x e. z) /\ z C_ (`'F"y)) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y))))
35	33, 34	bitri 190	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. z /\ z C_ (`'F"y)) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y))))
36	31, 35	syl6bb 595	. . . . . . . . . . . . . . . . . . . 20 |- ((F:X-->Y /\ z C_ dom F) -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y)))))
37	36	ex 402	. . . . . . . . . . . . . . . . . . 19 |- (F:X-->Y -> (z C_ dom F -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y))))))
38	6, 37	syld 30	. . . . . . . . . . . . . . . . . 18 |- (F:X-->Y -> (z e. J -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y))))))
39	38	imp 377	. . . . . . . . . . . . . . . . 17 |- ((F:X-->Y /\ z e. J) -> ((x e. z /\ (F"z) C_ y) <-> (x e. z /\ (x e. z /\ z C_ (`'F"y)))))
40	39	rexbidva 2120	. . . . . . . . . . . . . . . 16 |- (F:X-->Y -> (E.z e. J (x e. z /\ (F"z) C_ y) <-> E.z e. J (x e. z /\ (x e. z /\ z C_ (`'F"y)))))
41		elunirab 3190	. . . . . . . . . . . . . . . 16 |- (x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))} <-> E.z e. J (x e. z /\ (x e. z /\ z C_ (`'F"y))))
42	40, 41	syl6bbr 597	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> (E.z e. J (x e. z /\ (F"z) C_ y) <-> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}))
43	42	biimpd 170	. . . . . . . . . . . . . 14 |- (F:X-->Y -> (E.z e. J (x e. z /\ (F"z) C_ y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}))
44	43	adantr 425	. . . . . . . . . . . . 13 |- ((F:X-->Y /\ x e. dom F) -> (E.z e. J (x e. z /\ (F"z) C_ y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}))
45	30, 44	imim12d 69	. . . . . . . . . . . 12 |- ((F:X-->Y /\ x e. dom F) -> (((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (x e. (`'F"y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))})))
46	45	ex 402	. . . . . . . . . . 11 |- (F:X-->Y -> (x e. dom F -> (((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (x e. (`'F"y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}))))
47	46	com14 42	. . . . . . . . . 10 |- (x e. (`'F"y) -> (x e. dom F -> (((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (F:X-->Y -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}))))
48	27, 47	mpd 29	. . . . . . . . 9 |- (x e. (`'F"y) -> (((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (F:X-->Y -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))})))
49	48	com13 37	. . . . . . . 8 |- (F:X-->Y -> (((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (x e. (`'F"y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))})))
50	49	ralimdv 2172	. . . . . . 7 |- (F:X-->Y -> (A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> A.x e. X (x e. (`'F"y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))})))
51		cncnplem3 9053	. . . . . . 7 |- ((`'F"y) C_ X -> (A.x e. X (x e. (`'F"y) -> x e. U.{z e. J | (x e. z /\ z C_ (`'F"y))}) -> (`'F"y) C_ U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))}))
52	26, 50, 51	sylsyld 32	. . . . . 6 |- (F:X-->Y -> (A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y)) -> (`'F"y) C_ U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))}))
53	52	imp 377	. . . . 5 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> (`'F"y) C_ U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))})
54	23, 53	eqssd 2633	. . . 4 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ z C_ (`'F"y))} = (`'F"y))
55	18, 54	eqtrd 1925	. . 3 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} = (`'F"y))
56	55	eleq1d 1963	. 2 |- ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> (U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J <-> (`'F"y) e. J))
57		ssrab2 2692	. . . . . 6 |- {z e. J | (x e. z /\ (F"z) C_ y)} C_ J
58		uniopn 8867	. . . . . 6 |- ((J e. Top /\ {z e. J | (x e. z /\ (F"z) C_ y)} C_ J) -> U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J)
59	57, 58	mpan2 760	. . . . 5 |- (J e. Top -> U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J)
60	59	a1d 15	. . . 4 |- (J e. Top -> (x e. (`'F"y) -> U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J))
61	60	r19.21aiv 2175	. . 3 |- (J e. Top -> A.x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J)
62		iunopn 8868	. . 3 |- ((J e. Top /\ A.x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J) -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J)
63	61, 62	mpdan 768	. 2 |- (J e. Top -> U_x e. (`'F"y)U.{z e. J | (x e. z /\ (F"z) C_ y)} e. J)
64	56, 63	syl5cbi 226	1 |- (J e. Top -> ((F:X-->Y /\ A.x e. X ((F` x) e. y -> E.z e. J (x e. z /\ (F"z) C_ y))) -> (`'F"y) e. J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108   C_ wss 2593  U.cuni 3177  U_ciun 3255  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992  -->wf 3994  ` cfv 3998  Topctop 8857

This theorem is referenced by:  cncnp 9055

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-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-iun 3257  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-f 4010  df-fv 4014  df-top 8861

Copyright terms: Public domain