Theorem lmsslem 9230

Description: Lemma for lmss 9231 and causs 9233.

Hypotheses

Ref	Expression
lmsslem.1	|- (((D e. Met /\ ch) /\ ph) -> F C_ (CC X. dom dom D))
lmsslem.2	|- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F C_ (CC X. dom dom ( D |` (Y X. Y))))
lmsslem.3	|- ((D e. Met /\ (ch /\ F:NN-->dom dom ( D |` (Y X. Y)))) -> (ph <-> ps))
lmsslem.4	|- (ch -> F:NN-->Y)

Assertion

Ref	Expression
lmsslem	|- ((D e. Met /\ ch) -> (ph <-> ps))

Proof of Theorem lmsslem

Step	Hyp	Ref	Expression
1		lmsslem.1	. . . . 5 |- (((D e. Met /\ ch) /\ ph) -> F C_ (CC X. dom dom D))
2		sstr 2625	. . . . . 6 |- ((F C_ (CC X. dom dom ( D |` (Y X. Y))) /\ (CC X. dom dom ( D |` (Y X. Y))) C_ (CC X. dom dom D)) -> F C_ (CC X. dom dom D))
3		lmsslem.2	. . . . . . 7 |- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F C_ (CC X. dom dom ( D |` (Y X. Y))))
4		metres 9100	. . . . . . 7 |- (D e. Met -> (D |` (Y X. Y)) e. Met)
5	3, 4	sylanl1 509	. . . . . 6 |- (((D e. Met /\ ch) /\ ps) -> F C_ (CC X. dom dom ( D |` (Y X. Y))))
6		ssid 2634	. . . . . . 7 |- CC C_ CC
7		resss 4237	. . . . . . . . 9 |- (D |` (Y X. Y)) C_ D
8		dmss 4156	. . . . . . . . 9 |- ((D |` (Y X. Y)) C_ D -> dom ( D |` (Y X. Y)) C_ dom D)
9	7, 8	ax-mp 7	. . . . . . . 8 |- dom ( D |` (Y X. Y)) C_ dom D
10		dmss 4156	. . . . . . . 8 |- (dom ( D |` (Y X. Y)) C_ dom D -> dom dom ( D |` (Y X. Y)) C_ dom dom D)
11	9, 10	ax-mp 7	. . . . . . 7 |- dom dom ( D |` (Y X. Y)) C_ dom dom D
12		xpss12 4089	. . . . . . 7 |- ((CC C_ CC /\ dom dom ( D |` (Y X. Y)) C_ dom dom D) -> (CC X. dom dom ( D |` (Y X. Y))) C_ (CC X. dom dom D))
13	6, 11, 12	mp2an 761	. . . . . 6 |- (CC X. dom dom ( D |` (Y X. Y))) C_ (CC X. dom dom D)
14	2, 5, 13	sylancl 525	. . . . 5 |- (((D e. Met /\ ch) /\ ps) -> F C_ (CC X. dom dom D))
15	1, 14	jaodan 471	. . . 4 |- (((D e. Met /\ ch) /\ (ph \/ ps)) -> F C_ (CC X. dom dom D))
16	15	ex 402	. . 3 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> F C_ (CC X. dom dom D)))
17		simprl 450	. . . . . 6 |- ((D e. Met /\ (ch /\ F C_ (CC X. dom dom D))) -> ch)
18		eqid 1884	. . . . . . . . . . . 12 |- dom dom D = dom dom D
19	18	metssba 9086	. . . . . . . . . . 11 |- (D e. Met -> (dom dom D i^i Y) = dom dom ( D |` (Y X. Y)))
20		incom 2787	. . . . . . . . . . 11 |- (Y i^i dom dom D) = (dom dom D i^i Y)
21	19, 20	syl5req 1941	. . . . . . . . . 10 |- (D e. Met -> dom dom ( D |` (Y X. Y)) = (Y i^i dom dom D))
22		feq3 4553	. . . . . . . . . 10 |- (dom dom ( D |` (Y X. Y)) = (Y i^i dom dom D) -> (F:NN-->dom dom ( D |` (Y X. Y)) <-> F:NN-->(Y i^i dom dom D)))
23	21, 22	syl 12	. . . . . . . . 9 |- (D e. Met -> (F:NN-->dom dom ( D |` (Y X. Y)) <-> F:NN-->(Y i^i dom dom D)))
24		fin 4593	. . . . . . . . . 10 |- (F:NN-->(Y i^i dom dom D) <-> (F:NN-->Y /\ F:NN-->dom dom D))
25		simpl 346	. . . . . . . . . 10 |- ((F:NN-->Y /\ F C_ (CC X. dom dom D)) -> F:NN-->Y)
26		ffn 4562	. . . . . . . . . . . 12 |- (F:NN-->Y -> F Fn NN)
27		rnss 4189	. . . . . . . . . . . . 13 |- (F C_ (CC X. dom dom D) -> ran F C_ ran (CC X. dom dom D))
28		ax1cn 6422	. . . . . . . . . . . . . . 15 |- 1 e. CC
29		ne0i 2881	. . . . . . . . . . . . . . 15 |- (1 e. CC -> CC =/= (/))
30	28, 29	ax-mp 7	. . . . . . . . . . . . . 14 |- CC =/= (/)
31		rnxp 4342	. . . . . . . . . . . . . 14 |- (CC =/= (/) -> ran (CC X. dom dom D) = dom dom D)
32	30, 31	ax-mp 7	. . . . . . . . . . . . 13 |- ran (CC X. dom dom D) = dom dom D
33	27, 32	syl6ss 2663	. . . . . . . . . . . 12 |- (F C_ (CC X. dom dom D) -> ran F C_ dom dom D)
34	26, 33	anim12i 360	. . . . . . . . . . 11 |- ((F:NN-->Y /\ F C_ (CC X. dom dom D)) -> (F Fn NN /\ ran F C_ dom dom D))
35		df-f 4010	. . . . . . . . . . 11 |- (F:NN-->dom dom D <-> (F Fn NN /\ ran F C_ dom dom D))
36	34, 35	sylibr 217	. . . . . . . . . 10 |- ((F:NN-->Y /\ F C_ (CC X. dom dom D)) -> F:NN-->dom dom D)
37	24, 25, 36	sylanbrc 527	. . . . . . . . 9 |- ((F:NN-->Y /\ F C_ (CC X. dom dom D)) -> F:NN-->(Y i^i dom dom D))
38	23, 37	syl5bir 227	. . . . . . . 8 |- (D e. Met -> ((F:NN-->Y /\ F C_ (CC X. dom dom D)) -> F:NN-->dom dom ( D |` (Y X. Y))))
39	38	imp 377	. . . . . . 7 |- ((D e. Met /\ (F:NN-->Y /\ F C_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
40		lmsslem.4	. . . . . . 7 |- (ch -> F:NN-->Y)
41	39, 40	sylanr1 511	. . . . . 6 |- ((D e. Met /\ (ch /\ F C_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
42	17, 41	jca 310	. . . . 5 |- ((D e. Met /\ (ch /\ F C_ (CC X. dom dom D))) -> (ch /\ F:NN-->dom dom ( D |` (Y X. Y))))
43		lmsslem.3	. . . . 5 |- ((D e. Met /\ (ch /\ F:NN-->dom dom ( D |` (Y X. Y)))) -> (ph <-> ps))
44	42, 43	syldan 516	. . . 4 |- ((D e. Met /\ (ch /\ F C_ (CC X. dom dom D))) -> (ph <-> ps))
45	44	expr 418	. . 3 |- ((D e. Met /\ ch) -> (F C_ (CC X. dom dom D) -> (ph <-> ps)))
46	16, 45	syld 30	. 2 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> (ph <-> ps)))
47		oibabs 716	. 2 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
48	46, 47	sylib 215	1 |- ((D e. Met /\ ch) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   i^i cin 2592   C_ wss 2593  (/)c0 2875   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994  CCcc 6384  1c1 6387  NNcn 6449  Metcme 9066

This theorem is referenced by:  lmss 9231  causs 9233

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  ax-inf2 5731

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-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-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-enr 6318  df-nr 6319  df-0r 6323  df-1r 6324  df-c 6392  df-1 6394  df-r 6396  df-met 9070

Copyright terms: Public domain