Theorem aceq3lem 5894

Description: Lemma for aceq3 5895.

Hypothesis

Ref	Expression
aceq3lem.1	|- F = {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))}

Assertion

Ref	Expression
aceq3lem	|- (A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z) -> E.f(f C_ y /\ f Fn dom y))

Distinct variable group:   x,y,z,w,v,u,f

Proof of Theorem aceq3lem

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . 6 |- y e. _V
2	1	rnex 4209	. . . . 5 |- ran y e. _V
3	2	pwex 3487	. . . 4 |- ~Pran y e. _V
4		raleq 2266	. . . . 5 |- (x = ~Pran y -> (A.z e. x (z =/= (/) -> (f` z) e. z) <-> A.z e. ~P ran y(z =/= (/) -> (f` z) e. z)))
5	4	exbidv 1657	. . . 4 |- (x = ~Pran y -> (E.fA.z e. x (z =/= (/) -> (f` z) e. z) <-> E.fA.z e. ~P ran y(z =/= (/) -> (f` z) e. z)))
6	3, 5	cla4v 2370	. . 3 |- (A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z) -> E.fA.z e. ~P ran y(z =/= (/) -> (f` z) e. z))
7		fvex 4689	. . . . . . . 8 |- (f` {u | wyu}) e. _V
8		aceq3lem.1	. . . . . . . 8 |- F = {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))}
9	7, 8	fnopab2 4549	. . . . . . 7 |- F Fn dom y
10	1	dmex 4208	. . . . . . 7 |- dom y e. _V
11		fnex 4535	. . . . . . 7 |- ((F Fn dom y /\ dom y e. _V) -> F e. _V)
12	9, 10, 11	mp2an 761	. . . . . 6 |- F e. _V
13		sseq1 2637	. . . . . . 7 |- (g = F -> (g C_ y <-> F C_ y))
14		fneq1 4503	. . . . . . 7 |- (g = F -> (g Fn dom y <-> F Fn dom y))
15	13, 14	anbi12d 690	. . . . . 6 |- (g = F -> ((g C_ y /\ g Fn dom y) <-> (F C_ y /\ F Fn dom y)))
16	12, 15	cla4ev 2371	. . . . 5 |- ((F C_ y /\ F Fn dom y) -> E.g(g C_ y /\ g Fn dom y))
17		relopab 4104	. . . . . . . 8 |- Rel {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))}
18	8	releqi 4072	. . . . . . . 8 |- (Rel F <-> Rel {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))})
19	17, 18	mpbir 207	. . . . . . 7 |- Rel F
20	19	a1i 8	. . . . . 6 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> Rel F)
21	8	eleq2i 1961	. . . . . . . . . . 11 |- (<.t, h>. e. F <-> <.t, h>. e. {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))})
22		visset 2295	. . . . . . . . . . . 12 |- t e. _V
23		visset 2295	. . . . . . . . . . . 12 |- h e. _V
24		eleq1 1957	. . . . . . . . . . . . 13 |- (w = t -> (w e. dom y <-> t e. dom y))
25		breq1 3341	. . . . . . . . . . . . . . . 16 |- (w = t -> (wyu <-> tyu))
26	25	abbidv 2008	. . . . . . . . . . . . . . 15 |- (w = t -> {u | wyu} = {u | tyu})
27	26	fveq2d 4685	. . . . . . . . . . . . . 14 |- (w = t -> (f` {u | wyu}) = (f` {u | tyu}))
28	27	eqeq2d 1895	. . . . . . . . . . . . 13 |- (w = t -> (v = (f` {u | wyu}) <-> v = (f` {u | tyu})))
29	24, 28	anbi12d 690	. . . . . . . . . . . 12 |- (w = t -> ((w e. dom y /\ v = (f` {u | wyu})) <-> (t e. dom y /\ v = (f` {u | tyu}))))
30		eqeq1 1890	. . . . . . . . . . . . 13 |- (v = h -> (v = (f` {u | tyu}) <-> h = (f` {u | tyu})))
31	30	anbi2d 678	. . . . . . . . . . . 12 |- (v = h -> ((t e. dom y /\ v = (f` {u | tyu})) <-> (t e. dom y /\ h = (f` {u | tyu}))))
32	22, 23, 29, 31	opelopab 3570	. . . . . . . . . . 11 |- (<.t, h>. e. {<.w, v>. | (w e. dom y /\ v = (f` {u | wyu}))} <-> (t e. dom y /\ h = (f` {u | tyu})))
33	21, 32	bitri 190	. . . . . . . . . 10 |- (<.t, h>. e. F <-> (t e. dom y /\ h = (f` {u | tyu})))
34	33	simplbi 349	. . . . . . . . 9 |- (<.t, h>. e. F -> t e. dom y)
35		19.8a 1376	. . . . . . . . . . . . . . . 16 |- (tyu -> E.t tyu)
36	35	ss2abi 2679	. . . . . . . . . . . . . . 15 |- {u | tyu} C_ {u | E.t tyu}
37		dfrn2 4149	. . . . . . . . . . . . . . 15 |- ran y = {u | E.t tyu}
38	36, 37	sseqtr4i 2650	. . . . . . . . . . . . . 14 |- {u | tyu} C_ ran y
39	2	elpw2 3464	. . . . . . . . . . . . . 14 |- ({u | tyu} e. ~Pran y <-> {u | tyu} C_ ran y)
40	38, 39	mpbir 207	. . . . . . . . . . . . 13 |- {u | tyu} e. ~Pran y
41		neeq1 2024	. . . . . . . . . . . . . . 15 |- (z = {u | tyu} -> (z =/= (/) <-> {u | tyu} =/= (/)))
42		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (z = {u | tyu} -> (f` z) = (f` {u | tyu}))
43		id 73	. . . . . . . . . . . . . . . 16 |- (z = {u | tyu} -> z = {u | tyu})
44	42, 43	eleq12d 1965	. . . . . . . . . . . . . . 15 |- (z = {u | tyu} -> ((f` z) e. z <-> (f` {u | tyu}) e. {u | tyu}))
45	41, 44	imbi12d 688	. . . . . . . . . . . . . 14 |- (z = {u | tyu} -> ((z =/= (/) -> (f` z) e. z) <-> ({u | tyu} =/= (/) -> (f` {u | tyu}) e. {u | tyu})))
46	45	rcla4v 2376	. . . . . . . . . . . . 13 |- ({u | tyu} e. ~Pran y -> (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> ({u | tyu} =/= (/) -> (f` {u | tyu}) e. {u | tyu})))
47	40, 46	ax-mp 7	. . . . . . . . . . . 12 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> ({u | tyu} =/= (/) -> (f` {u | tyu}) e. {u | tyu}))
48	22	eldm 4153	. . . . . . . . . . . . 13 |- (t e. dom y <-> E.u tyu)
49		abn0 2892	. . . . . . . . . . . . 13 |- ({u | tyu} =/= (/) <-> E.u tyu)
50	48, 49	bitr4i 193	. . . . . . . . . . . 12 |- (t e. dom y <-> {u | tyu} =/= (/))
51	47, 50	syl5ib 223	. . . . . . . . . . 11 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> (t e. dom y -> (f` {u | tyu}) e. {u | tyu}))
52	51	com12 14	. . . . . . . . . 10 |- (t e. dom y -> (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> (f` {u | tyu}) e. {u | tyu}))
53		fvex 4689	. . . . . . . . . . . . 13 |- (f` {u | tyu}) e. _V
54	27, 8, 53	fvopab4 4743	. . . . . . . . . . . 12 |- (t e. dom y -> (F` t) = (f` {u | tyu}))
55	54	eleq1d 1963	. . . . . . . . . . 11 |- (t e. dom y -> ((F` t) e. {u | tyu} <-> (f` {u | tyu}) e. {u | tyu}))
56		fvex 4689	. . . . . . . . . . . . 13 |- (F` t) e. _V
57		breq2 3342	. . . . . . . . . . . . 13 |- (h = (F` t) -> (tyh <-> ty(F` t)))
58		breq2 3342	. . . . . . . . . . . . . 14 |- (u = h -> (tyu <-> tyh))
59	58	cbvabv 2420	. . . . . . . . . . . . 13 |- {u | tyu} = {h | tyh}
60	56, 57, 59	elab2 2407	. . . . . . . . . . . 12 |- ((F` t) e. {u | tyu} <-> ty(F` t))
61		df-br 3339	. . . . . . . . . . . 12 |- (ty(F` t) <-> <.t, (F` t)>. e. y)
62	60, 61	bitr2i 191	. . . . . . . . . . 11 |- (<.t, (F` t)>. e. y <-> (F` t) e. {u | tyu})
63	55, 62	syl5bb 591	. . . . . . . . . 10 |- (t e. dom y -> (<.t, (F` t)>. e. y <-> (f` {u | tyu}) e. {u | tyu}))
64	52, 63	sylibrd 221	. . . . . . . . 9 |- (t e. dom y -> (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> <.t, (F` t)>. e. y))
65	34, 64	syl 12	. . . . . . . 8 |- (<.t, h>. e. F -> (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> <.t, (F` t)>. e. y))
66		fnfun 4510	. . . . . . . . . . 11 |- (F Fn dom y -> Fun F)
67	9, 66	ax-mp 7	. . . . . . . . . 10 |- Fun F
68	23	funopfv 4710	. . . . . . . . . 10 |- (Fun F -> (<.t, h>. e. F -> (F` t) = h))
69	67, 68	ax-mp 7	. . . . . . . . 9 |- (<.t, h>. e. F -> (F` t) = h)
70		opeq2 3159	. . . . . . . . . 10 |- ((F` t) = h -> <.t, (F` t)>. = <.t, h>.)
71	70	eleq1d 1963	. . . . . . . . 9 |- ((F` t) = h -> (<.t, (F` t)>. e. y <-> <.t, h>. e. y))
72	69, 71	syl 12	. . . . . . . 8 |- (<.t, h>. e. F -> (<.t, (F` t)>. e. y <-> <.t, h>. e. y))
73	65, 72	sylibd 219	. . . . . . 7 |- (<.t, h>. e. F -> (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> <.t, h>. e. y))
74	73	com12 14	. . . . . 6 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> (<.t, h>. e. F -> <.t, h>. e. y))
75	20, 74	relssdv 4079	. . . . 5 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> F C_ y)
76	16, 75, 9	sylancl 525	. . . 4 |- (A.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> E.g(g C_ y /\ g Fn dom y))
77	76	19.23aiv 1674	. . 3 |- (E.fA.z e. ~P ran y(z =/= (/) -> (f` z) e. z) -> E.g(g C_ y /\ g Fn dom y))
78	6, 77	syl 12	. 2 |- (A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z) -> E.g(g C_ y /\ g Fn dom y))
79		sseq1 2637	. . . 4 |- (g = f -> (g C_ y <-> f C_ y))
80		fneq1 4503	. . . 4 |- (g = f -> (g Fn dom y <-> f Fn dom y))
81	79, 80	anbi12d 690	. . 3 |- (g = f -> ((g C_ y /\ g Fn dom y) <-> (f C_ y /\ f Fn dom y)))
82	81	cbvexv 1697	. 2 |- (E.g(g C_ y /\ g Fn dom y) <-> E.f(f C_ y /\ f Fn dom y))
83	78, 82	sylib 215	1 |- (A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z) -> E.f(f C_ y /\ f Fn dom y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  <.cop 3046   class class class wbr 3338  {copab 3395  dom cdm 3986  ran crn 3987  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  aceq3 5895

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-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-v 2294  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

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