Theorem fnctartar 15284

Description: Consider functions whose domain A is an element of a transitive Tarski's class T and whose range is T, then they are elements of T. CLASSES2 th. 23.

Assertion

Ref	Expression
fnctartar	|- ((T e. Tarski /\ Tr T /\ A e. T) -> (T ^m A) C_ T)

Proof of Theorem fnctartar

Step	Hyp	Ref	Expression
1		elmapg 5392	. . . . . 6 |- ((T e. Tarski /\ A e. T) -> (f e. (T ^m A) <-> f:A-->T))
2		simpr1 882	. . . . . . . . 9 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> T e. Tarski )
3		fssxp 4575	. . . . . . . . . . . . 13 |- (f:A-->T -> f C_ (A X. T))
4		ssel 2615	. . . . . . . . . . . . . . 15 |- (f C_ (A X. T) -> (x e. f -> x e. (A X. T)))
5		elxp6 5041	. . . . . . . . . . . . . . . 16 |- (x e. (A X. T) <-> (x = <.(1st` x), (2nd` x)>. /\ ((1st` x) e. A /\ (2nd` x) e. T)))
6		eleq1 1957	. . . . . . . . . . . . . . . . . . . 20 |- (x = <.(1st` x), (2nd` x)>. -> (x e. T <-> <.(1st` x), (2nd` x)>. e. T))
7	6	imbi2d 674	. . . . . . . . . . . . . . . . . . 19 |- (x = <.(1st` x), (2nd` x)>. -> (((T e. Tarski /\ Tr T /\ A e. T) -> x e. T) <-> ((T e. Tarski /\ Tr T /\ A e. T) -> <.(1st` x), (2nd` x)>. e. T)))
8	7	imbi2d 674	. . . . . . . . . . . . . . . . . 18 |- (x = <.(1st` x), (2nd` x)>. -> ((f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> x e. T)) <-> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> <.(1st` x), (2nd` x)>. e. T))))
9		simp31 912	. . . . . . . . . . . . . . . . . . . 20 |- ((((1st` x) e. A /\ (2nd` x) e. T) /\ f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> T e. Tarski )
10		trel 3418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (Tr T -> (((1st` x) e. A /\ A e. T) -> (1st` x) e. T))
11	10	exp3a 405	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (Tr T -> ((1st` x) e. A -> (A e. T -> (1st` x) e. T)))
12	11	com23 36	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (Tr T -> (A e. T -> ((1st` x) e. A -> (1st` x) e. T)))
13	12	imp 377	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Tr T /\ A e. T) -> ((1st` x) e. A -> (1st` x) e. T))
14	13	3adant1 894	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((T e. Tarski /\ Tr T /\ A e. T) -> ((1st` x) e. A -> (1st` x) e. T))
15	14	com12 14	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((1st` x) e. A -> ((T e. Tarski /\ Tr T /\ A e. T) -> (1st` x) e. T))
16	15	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- (((1st` x) e. A /\ (2nd` x) e. T) -> ((T e. Tarski /\ Tr T /\ A e. T) -> (1st` x) e. T))
17	16	imp 377	. . . . . . . . . . . . . . . . . . . . 21 |- ((((1st` x) e. A /\ (2nd` x) e. T) /\ (T e. Tarski /\ Tr T /\ A e. T)) -> (1st` x) e. T)
18	17	3adant2 895	. . . . . . . . . . . . . . . . . . . 20 |- ((((1st` x) e. A /\ (2nd` x) e. T) /\ f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> (1st` x) e. T)
19		simp1r 901	. . . . . . . . . . . . . . . . . . . 20 |- ((((1st` x) e. A /\ (2nd` x) e. T) /\ f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> (2nd` x) e. T)
20		tarorpa 15236	. . . . . . . . . . . . . . . . . . . 20 |- ((T e. Tarski /\ (1st` x) e. T /\ (2nd` x) e. T) -> <.(1st` x), (2nd` x)>. e. T)
21	9, 18, 19, 20	syl111anc 1100	. . . . . . . . . . . . . . . . . . 19 |- ((((1st` x) e. A /\ (2nd` x) e. T) /\ f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> <.(1st` x), (2nd` x)>. e. T)
22	21	3exp 1066	. . . . . . . . . . . . . . . . . 18 |- (((1st` x) e. A /\ (2nd` x) e. T) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> <.(1st` x), (2nd` x)>. e. T)))
23	8, 22	syl5bir 227	. . . . . . . . . . . . . . . . 17 |- (x = <.(1st` x), (2nd` x)>. -> (((1st` x) e. A /\ (2nd` x) e. T) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> x e. T))))
24	23	imp 377	. . . . . . . . . . . . . . . 16 |- ((x = <.(1st` x), (2nd` x)>. /\ ((1st` x) e. A /\ (2nd` x) e. T)) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> x e. T)))
25	5, 24	sylbi 216	. . . . . . . . . . . . . . 15 |- (x e. (A X. T) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> x e. T)))
26	4, 25	syl6com 64	. . . . . . . . . . . . . 14 |- (x e. f -> (f C_ (A X. T) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> x e. T))))
27	26	com4l 43	. . . . . . . . . . . . 13 |- (f C_ (A X. T) -> (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> (x e. f -> x e. T))))
28	3, 27	mpcom 60	. . . . . . . . . . . 12 |- (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> (x e. f -> x e. T)))
29	28	imp 377	. . . . . . . . . . 11 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> (x e. f -> x e. T))
30	29	19.21aiv 1664	. . . . . . . . . 10 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> A.x(x e. f -> x e. T))
31		dfss2 2610	. . . . . . . . . 10 |- (f C_ T <-> A.x(x e. f -> x e. T))
32	30, 31	sylibr 217	. . . . . . . . 9 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> f C_ T)
33		ffn 4562	. . . . . . . . . . 11 |- (f:A-->T -> f Fn A)
34		fnfun 4510	. . . . . . . . . . . 12 |- (f Fn A -> Fun f)
35		fndm 4512	. . . . . . . . . . . 12 |- (f Fn A -> dom f = A)
36	34, 35	jca 310	. . . . . . . . . . 11 |- (f Fn A -> (Fun f /\ dom f = A))
37		visset 2295	. . . . . . . . . . . . 13 |- f e. _V
38	37	fundmen 5487	. . . . . . . . . . . 12 |- (Fun f -> dom f ~~ f)
39		breq1 3341	. . . . . . . . . . . . 13 |- (dom f = A -> (dom f ~~ f <-> A ~~ f))
40		sdomen1 5544	. . . . . . . . . . . . . . . . . 18 |- ((f e. _V /\ A ~~ f) -> (A ~< T <-> f ~< T))
41	40	biimpd 170	. . . . . . . . . . . . . . . . 17 |- ((f e. _V /\ A ~~ f) -> (A ~< T -> f ~< T))
42	37, 41	mpan 759	. . . . . . . . . . . . . . . 16 |- (A ~~ f -> (A ~< T -> f ~< T))
43		tarax3d2 15225	. . . . . . . . . . . . . . . 16 |- ((T e. Tarski /\ A e. T) -> A ~< T)
44	42, 43	syl5com 63	. . . . . . . . . . . . . . 15 |- ((T e. Tarski /\ A e. T) -> (A ~~ f -> f ~< T))
45	44	3adant2 895	. . . . . . . . . . . . . 14 |- ((T e. Tarski /\ Tr T /\ A e. T) -> (A ~~ f -> f ~< T))
46	45	com12 14	. . . . . . . . . . . . 13 |- (A ~~ f -> ((T e. Tarski /\ Tr T /\ A e. T) -> f ~< T))
47	39, 46	syl6bi 231	. . . . . . . . . . . 12 |- (dom f = A -> (dom f ~~ f -> ((T e. Tarski /\ Tr T /\ A e. T) -> f ~< T)))
48	38, 47	mpan9 521	. . . . . . . . . . 11 |- ((Fun f /\ dom f = A) -> ((T e. Tarski /\ Tr T /\ A e. T) -> f ~< T))
49	33, 36, 48	3syl 24	. . . . . . . . . 10 |- (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> f ~< T))
50	49	imp 377	. . . . . . . . 9 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> f ~< T)
51	2, 32, 50	3jca 1050	. . . . . . . 8 |- ((f:A-->T /\ (T e. Tarski /\ Tr T /\ A e. T)) -> (T e. Tarski /\ f C_ T /\ f ~< T))
52	51	ex 402	. . . . . . 7 |- (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> (T e. Tarski /\ f C_ T /\ f ~< T)))
53		tarax3f 15229	. . . . . . 7 |- ((T e. Tarski /\ f C_ T /\ f ~< T) -> f e. T)
54	52, 53	syl6 25	. . . . . 6 |- (f:A-->T -> ((T e. Tarski /\ Tr T /\ A e. T) -> f e. T))
55	1, 54	syl6bi 231	. . . . 5 |- ((T e. Tarski /\ A e. T) -> (f e. (T ^m A) -> ((T e. Tarski /\ Tr T /\ A e. T) -> f e. T)))
56	55	3adant2 895	. . . 4 |- ((T e. Tarski /\ Tr T /\ A e. T) -> (f e. (T ^m A) -> ((T e. Tarski /\ Tr T /\ A e. T) -> f e. T)))
57	56	pm2.43a 80	. . 3 |- ((T e. Tarski /\ Tr T /\ A e. T) -> (f e. (T ^m A) -> f e. T))
58	57	19.21aiv 1664	. 2 |- ((T e. Tarski /\ Tr T /\ A e. T) -> A.f(f e. (T ^m A) -> f e. T))
59		dfss2 2610	. 2 |- ((T ^m A) C_ T <-> A.f(f e. (T ^m A) -> f e. T))
60	58, 59	sylibr 217	1 |- ((T e. Tarski /\ Tr T /\ A e. T) -> (T ^m A) C_ T)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046   class class class wbr 3338  Tr wtr 3411   X. cxp 3984  dom cdm 3986  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019   ^m cmap 5381   ~~ cen 5423   ~< csdm 5425   Tarski ctarski 15208

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-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-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  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-suc 3663  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-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-1o 5177  df-2o 5178  df-er 5318  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-tsk 15210

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