Theorem lmsslem 7952

Description: Lemma for lmss 7953 and causs 7955.

Hypotheses

Ref	Expression
lmsslem.1	|- (((D e. Met /\ ch) /\ ph) -> F (_ (CC X. dom dom D))
lmsslem.2	|- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F (_ (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 (_ (CC X. dom dom D))
2		lmsslem.2	. . . . . . 7 |- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom ( D |` (Y X. Y))))
3		metres 7823	. . . . . . 7 |- (D e. Met -> (D |` (Y X. Y)) e. Met)
4	2, 3	sylanl1 460	. . . . . 6 |- (((D e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom ( D |` (Y X. Y))))
5		ssid 2080	. . . . . . . 8 |- CC (_ CC
6		resss 3383	. . . . . . . . . 10 |- (D |` (Y X. Y)) (_ D
7		dmss 3310	. . . . . . . . . 10 |- ((D |` (Y X. Y)) (_ D -> dom ( D |` (Y X. Y)) (_ dom D)
8	6, 7	ax-mp 7	. . . . . . . . 9 |- dom ( D |` (Y X. Y)) (_ dom D
9		dmss 3310	. . . . . . . . 9 |- (dom ( D |` (Y X. Y)) (_ dom D -> dom dom ( D |` (Y X. Y)) (_ dom dom D)
10	8, 9	ax-mp 7	. . . . . . . 8 |- dom dom ( D |` (Y X. Y)) (_ dom dom D
11		ssxp 3256	. . . . . . . 8 |- ((CC (_ CC /\ dom dom ( D |` (Y X. Y)) (_ dom dom D) -> (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D))
12	5, 10, 11	mp2an 697	. . . . . . 7 |- (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D)
13		sstr 2072	. . . . . . 7 |- ((F (_ (CC X. dom dom ( D |` (Y X. Y))) /\ (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D)) -> F (_ (CC X. dom dom D))
14	12, 13	mpan2 696	. . . . . 6 |- (F (_ (CC X. dom dom ( D |` (Y X. Y))) -> F (_ (CC X. dom dom D))
15	4, 14	syl 10	. . . . 5 |- (((D e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom D))
16	1, 15	jaodan 426	. . . 4 |- (((D e. Met /\ ch) /\ (ph \/ ps)) -> F (_ (CC X. dom dom D))
17	16	ex 373	. . 3 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> F (_ (CC X. dom dom D)))
18		simprl 414	. . . . . . 7 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> ch)
19		eqid 1475	. . . . . . . . . . . . 13 |- dom dom D = dom dom D
20	19	metssba 7809	. . . . . . . . . . . 12 |- (D e. Met -> (dom dom D i^i Y) = dom dom ( D |` (Y X. Y)))
21		incom 2208	. . . . . . . . . . . 12 |- (Y i^i dom dom D) = (dom dom D i^i Y)
22	20, 21	syl5req 1520	. . . . . . . . . . 11 |- (D e. Met -> dom dom ( D |` (Y X. Y)) = (Y i^i dom dom D))
23		feq3 3622	. . . . . . . . . . 11 |- (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)))
24	22, 23	syl 10	. . . . . . . . . 10 |- (D e. Met -> (F:NN-->dom dom ( D |` (Y X. Y)) <-> F:NN-->(Y i^i dom dom D)))
25		pm3.26 319	. . . . . . . . . . . 12 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->Y)
26		ffn 3627	. . . . . . . . . . . . . 14 |- (F:NN-->Y -> F Fn NN)
27		rnss 3342	. . . . . . . . . . . . . . 15 |- (F (_ (CC X. dom dom D) -> ran F (_ ran (CC X. dom dom D))
28		ax1cn 5269	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
29		ne0i 2286	. . . . . . . . . . . . . . . . 17 |- (1 e. CC -> CC =/= (/))
30	28, 29	ax-mp 7	. . . . . . . . . . . . . . . 16 |- CC =/= (/)
31		rnxp 3472	. . . . . . . . . . . . . . . 16 |- (CC =/= (/) -> ran (CC X. dom dom D) = dom dom D)
32	30, 31	ax-mp 7	. . . . . . . . . . . . . . 15 |- ran (CC X. dom dom D) = dom dom D
33	27, 32	syl6ss 2107	. . . . . . . . . . . . . 14 |- (F (_ (CC X. dom dom D) -> ran F (_ dom dom D)
34	26, 33	anim12i 333	. . . . . . . . . . . . 13 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> (F Fn NN /\ ran F (_ dom dom D))
35		df-f 3194	. . . . . . . . . . . . 13 |- (F:NN-->dom dom D <-> (F Fn NN /\ ran F (_ dom dom D))
36	34, 35	sylibr 200	. . . . . . . . . . . 12 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->dom dom D)
37	25, 36	jca 288	. . . . . . . . . . 11 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> (F:NN-->Y /\ F:NN-->dom dom D))
38		fin 3651	. . . . . . . . . . 11 |- (F:NN-->(Y i^i dom dom D) <-> (F:NN-->Y /\ F:NN-->dom dom D))
39	37, 38	sylibr 200	. . . . . . . . . 10 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->(Y i^i dom dom D))
40	24, 39	syl5bir 210	. . . . . . . . 9 |- (D e. Met -> ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->dom dom ( D |` (Y X. Y))))
41	40	imp 350	. . . . . . . 8 |- ((D e. Met /\ (F:NN-->Y /\ F (_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
42		lmsslem.4	. . . . . . . 8 |- (ch -> F:NN-->Y)
43	41, 42	sylanr1 462	. . . . . . 7 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
44	18, 43	jca 288	. . . . . 6 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> (ch /\ F:NN-->dom dom ( D |` (Y X. Y))))
45		lmsslem.3	. . . . . 6 |- ((D e. Met /\ (ch /\ F:NN-->dom dom ( D |` (Y X. Y)))) -> (ph <-> ps))
46	44, 45	syldan 467	. . . . 5 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> (ph <-> ps))
47	46	anassrs 441	. . . 4 |- (((D e. Met /\ ch) /\ F (_ (CC X. dom dom D)) -> (ph <-> ps))
48	47	ex 373	. . 3 |- ((D e. Met /\ ch) -> (F (_ (CC X. dom dom D) -> (ph <-> ps)))
49	17, 48	syld 27	. 2 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> (ph <-> ps)))
50		oibabs 654	. 2 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
51	49, 50	sylib 198	1 |- ((D e. Met /\ ch) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   i^i cin 2046   (_ wss 2047  (/)c0 2280   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172   Fn wfn 3177  -->wf 3178  CCcc 5232  1c1 5235  NNcn 5296  Metcme 7789

This theorem is referenced by:  lmss 7953  causs 7955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-enr 5166  df-nr 5167  df-0r 5171  df-1r 5172  df-c 5240  df-1 5242  df-r 5244  df-met 7793

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