Theorem isofrlem 4878

Description: Lemma for isofr 4879.

Assertion

Ref	Expression
isofrlem	|- (H Isom R, S (A, B) -> (S Fr B -> R Fr A))

Proof of Theorem isofrlem

Step	Hyp	Ref	Expression
1		isof1o 4870	. . . . . . 7 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
2		f1ofun 4637	. . . . . . 7 |- (H:A-1-1-onto->B -> Fun H)
3		visset 2295	. . . . . . . . 9 |- x e. _V
4	3	funimaex 4496	. . . . . . . 8 |- (Fun H -> (H"x) e. _V)
5		sseq1 2637	. . . . . . . . . . 11 |- (z = (H"x) -> (z C_ B <-> (H"x) C_ B))
6		neeq1 2024	. . . . . . . . . . 11 |- (z = (H"x) -> (z =/= (/) <-> (H"x) =/= (/)))
7	5, 6	anbi12d 690	. . . . . . . . . 10 |- (z = (H"x) -> ((z C_ B /\ z =/= (/)) <-> ((H"x) C_ B /\ (H"x) =/= (/))))
8		ineq1 2789	. . . . . . . . . . . 12 |- (z = (H"x) -> (z i^i (`'S"{w})) = ((H"x) i^i (`'S"{w})))
9	8	eqeq1d 1892	. . . . . . . . . . 11 |- (z = (H"x) -> ((z i^i (`'S"{w})) = (/) <-> ((H"x) i^i (`'S"{w})) = (/)))
10	9	rexeqbi1dv 2272	. . . . . . . . . 10 |- (z = (H"x) -> (E.w e. z (z i^i (`'S"{w})) = (/) <-> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/)))
11	7, 10	imbi12d 688	. . . . . . . . 9 |- (z = (H"x) -> (((z C_ B /\ z =/= (/)) -> E.w e. z (z i^i (`'S"{w})) = (/)) <-> (((H"x) C_ B /\ (H"x) =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
12	11	cla4gv 2364	. . . . . . . 8 |- ((H"x) e. _V -> (A.z((z C_ B /\ z =/= (/)) -> E.w e. z (z i^i (`'S"{w})) = (/)) -> (((H"x) C_ B /\ (H"x) =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
13	4, 12	syl 12	. . . . . . 7 |- (Fun H -> (A.z((z C_ B /\ z =/= (/)) -> E.w e. z (z i^i (`'S"{w})) = (/)) -> (((H"x) C_ B /\ (H"x) =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
14	1, 2, 13	3syl 24	. . . . . 6 |- (H Isom R, S (A, B) -> (A.z((z C_ B /\ z =/= (/)) -> E.w e. z (z i^i (`'S"{w})) = (/)) -> (((H"x) C_ B /\ (H"x) =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
15		dffr3 4297	. . . . . 6 |- (S Fr B <-> A.z((z C_ B /\ z =/= (/)) -> E.w e. z (z i^i (`'S"{w})) = (/)))
16	14, 15	syl5ib 223	. . . . 5 |- (H Isom R, S (A, B) -> (S Fr B -> (((H"x) C_ B /\ (H"x) =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
17		f1ofn 4636	. . . . . . . 8 |- (H:A-1-1-onto->B -> H Fn A)
18		ssel 2615	. . . . . . . . . . . . 13 |- (x C_ A -> (y e. x -> y e. A))
19		funfvima 4828	. . . . . . . . . . . . . . . 16 |- ((Fun H /\ y e. dom H) -> (y e. x -> (H` y) e. (H"x)))
20	19	funfni 4513	. . . . . . . . . . . . . . 15 |- ((H Fn A /\ y e. A) -> (y e. x -> (H` y) e. (H"x)))
21		ne0i 2881	. . . . . . . . . . . . . . 15 |- ((H` y) e. (H"x) -> (H"x) =/= (/))
22	20, 21	syl6 25	. . . . . . . . . . . . . 14 |- ((H Fn A /\ y e. A) -> (y e. x -> (H"x) =/= (/)))
23	22	ex 402	. . . . . . . . . . . . 13 |- (H Fn A -> (y e. A -> (y e. x -> (H"x) =/= (/))))
24	18, 23	sylan9r 519	. . . . . . . . . . . 12 |- ((H Fn A /\ x C_ A) -> (y e. x -> (y e. x -> (H"x) =/= (/))))
25	24	pm2.43d 79	. . . . . . . . . . 11 |- ((H Fn A /\ x C_ A) -> (y e. x -> (H"x) =/= (/)))
26	25	19.23adv 1584	. . . . . . . . . 10 |- ((H Fn A /\ x C_ A) -> (E.y y e. x -> (H"x) =/= (/)))
27		n0 2884	. . . . . . . . . 10 |- (x =/= (/) <-> E.y y e. x)
28	26, 27	syl5ib 223	. . . . . . . . 9 |- ((H Fn A /\ x C_ A) -> (x =/= (/) -> (H"x) =/= (/)))
29	28	expimpd 404	. . . . . . . 8 |- (H Fn A -> ((x C_ A /\ x =/= (/)) -> (H"x) =/= (/)))
30	17, 29	syl 12	. . . . . . 7 |- (H:A-1-1-onto->B -> ((x C_ A /\ x =/= (/)) -> (H"x) =/= (/)))
31		f1ofo 4643	. . . . . . . 8 |- (H:A-1-1-onto->B -> H:A-onto->B)
32		imassrn 4278	. . . . . . . . 9 |- (H"x) C_ ran H
33		forn 4620	. . . . . . . . . 10 |- (H:A-onto->B -> ran H = B)
34	33	sseq2d 2645	. . . . . . . . 9 |- (H:A-onto->B -> ((H"x) C_ ran H <-> (H"x) C_ B))
35	32, 34	mpbii 210	. . . . . . . 8 |- (H:A-onto->B -> (H"x) C_ B)
36	31, 35	syl 12	. . . . . . 7 |- (H:A-1-1-onto->B -> (H"x) C_ B)
37	30, 36	jctild 662	. . . . . 6 |- (H:A-1-1-onto->B -> ((x C_ A /\ x =/= (/)) -> ((H"x) C_ B /\ (H"x) =/= (/))))
38	1, 37	syl 12	. . . . 5 |- (H Isom R, S (A, B) -> ((x C_ A /\ x =/= (/)) -> ((H"x) C_ B /\ (H"x) =/= (/))))
39	16, 38	syl5d 66	. . . 4 |- (H Isom R, S (A, B) -> (S Fr B -> ((x C_ A /\ x =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/))))
40		fvelima 4723	. . . . . . . . . . 11 |- ((Fun H /\ w e. (H"x)) -> E.y e. x (H` y) = w)
41	1	adantr 425	. . . . . . . . . . . 12 |- ((H Isom R, S (A, B) /\ x C_ A) -> H:A-1-1-onto->B)
42	41, 2	syl 12	. . . . . . . . . . 11 |- ((H Isom R, S (A, B) /\ x C_ A) -> Fun H)
43		simpl 346	. . . . . . . . . . 11 |- ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> w e. (H"x))
44	40, 42, 43	syl2an 503	. . . . . . . . . 10 |- (((H Isom R, S (A, B) /\ x C_ A) /\ (w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/))) -> E.y e. x (H` y) = w)
45		isomin 4876	. . . . . . . . . . . . . . . . . . 19 |- ((H Isom R, S (A, B) /\ (x C_ A /\ y e. A)) -> ((x i^i (`'R"{y})) = (/) <-> ((H"x) i^i (`'S"{(H` y)})) = (/)))
46	18	imdistani 491	. . . . . . . . . . . . . . . . . . 19 |- ((x C_ A /\ y e. x) -> (x C_ A /\ y e. A))
47	45, 46	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- ((H Isom R, S (A, B) /\ (x C_ A /\ y e. x)) -> ((x i^i (`'R"{y})) = (/) <-> ((H"x) i^i (`'S"{(H` y)})) = (/)))
48		sneq 3054	. . . . . . . . . . . . . . . . . . . . 21 |- ((H` y) = w -> {(H` y)} = {w})
49	48	imaeq2d 4264	. . . . . . . . . . . . . . . . . . . 20 |- ((H` y) = w -> (`'S"{(H` y)}) = (`'S"{w}))
50	49	ineq2d 2796	. . . . . . . . . . . . . . . . . . 19 |- ((H` y) = w -> ((H"x) i^i (`'S"{(H` y)})) = ((H"x) i^i (`'S"{w})))
51	50	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- ((H` y) = w -> (((H"x) i^i (`'S"{(H` y)})) = (/) <-> ((H"x) i^i (`'S"{w})) = (/)))
52	47, 51	sylan9bb 599	. . . . . . . . . . . . . . . . 17 |- (((H Isom R, S (A, B) /\ (x C_ A /\ y e. x)) /\ (H` y) = w) -> ((x i^i (`'R"{y})) = (/) <-> ((H"x) i^i (`'S"{w})) = (/)))
53		simpr 350	. . . . . . . . . . . . . . . . 17 |- ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> ((H"x) i^i (`'S"{w})) = (/))
54	52, 53	syl5bir 227	. . . . . . . . . . . . . . . 16 |- (((H Isom R, S (A, B) /\ (x C_ A /\ y e. x)) /\ (H` y) = w) -> ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> (x i^i (`'R"{y})) = (/)))
55	54	exp42 414	. . . . . . . . . . . . . . 15 |- (H Isom R, S (A, B) -> (x C_ A -> (y e. x -> ((H` y) = w -> ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> (x i^i (`'R"{y})) = (/))))))
56	55	imp 377	. . . . . . . . . . . . . 14 |- ((H Isom R, S (A, B) /\ x C_ A) -> (y e. x -> ((H` y) = w -> ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> (x i^i (`'R"{y})) = (/)))))
57	56	com3l 38	. . . . . . . . . . . . 13 |- (y e. x -> ((H` y) = w -> ((H Isom R, S (A, B) /\ x C_ A) -> ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> (x i^i (`'R"{y})) = (/)))))
58	57	com4t 44	. . . . . . . . . . . 12 |- ((H Isom R, S (A, B) /\ x C_ A) -> ((w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/)) -> (y e. x -> ((H` y) = w -> (x i^i (`'R"{y})) = (/)))))
59	58	imp 377	. . . . . . . . . . 11 |- (((H Isom R, S (A, B) /\ x C_ A) /\ (w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/))) -> (y e. x -> ((H` y) = w -> (x i^i (`'R"{y})) = (/))))
60	59	reximdvai 2201	. . . . . . . . . 10 |- (((H Isom R, S (A, B) /\ x C_ A) /\ (w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/))) -> (E.y e. x (H` y) = w -> E.y e. x (x i^i (`'R"{y})) = (/)))
61	44, 60	mpd 29	. . . . . . . . 9 |- (((H Isom R, S (A, B) /\ x C_ A) /\ (w e. (H"x) /\ ((H"x) i^i (`'S"{w})) = (/))) -> E.y e. x (x i^i (`'R"{y})) = (/))
62	61	exp32 408	. . . . . . . 8 |- ((H Isom R, S (A, B) /\ x C_ A) -> (w e. (H"x) -> (((H"x) i^i (`'S"{w})) = (/) -> E.y e. x (x i^i (`'R"{y})) = (/))))
63	62	r19.23adv 2215	. . . . . . 7 |- ((H Isom R, S (A, B) /\ x C_ A) -> (E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/) -> E.y e. x (x i^i (`'R"{y})) = (/)))
64	63	ex 402	. . . . . 6 |- (H Isom R, S (A, B) -> (x C_ A -> (E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/) -> E.y e. x (x i^i (`'R"{y})) = (/))))
65	64	adantrd 427	. . . . 5 |- (H Isom R, S (A, B) -> ((x C_ A /\ x =/= (/)) -> (E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/) -> E.y e. x (x i^i (`'R"{y})) = (/))))
66	65	a2d 16	. . . 4 |- (H Isom R, S (A, B) -> (((x C_ A /\ x =/= (/)) -> E.w e. (H"x)((H"x) i^i (`'S"{w})) = (/)) -> ((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/))))
67	39, 66	syld 30	. . 3 |- (H Isom R, S (A, B) -> (S Fr B -> ((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/))))
68	67	19.21adv 1666	. 2 |- (H Isom R, S (A, B) -> (S Fr B -> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/))))
69		dffr3 4297	. 2 |- (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))
70	68, 69	syl6ibr 230	1 |- (H Isom R, S (A, B) -> (S Fr B -> R Fr A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   Fr wfr 3623  `'ccnv 3985  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998   Isom wiso 3999

This theorem is referenced by:  isofr 4879

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-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-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-fr 3625  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015

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