Theorem trnij 15015

Description: A translation is 1-1-onto.

Hypothesis

Ref	Expression
trnij.1	|- F = (x e. RR |-> (x + A))

Assertion

Ref	Expression
trnij	|- (A e. RR -> F:RR-1-1-onto->RR)

Distinct variable groups:   x,A   x,F

Proof of Theorem trnij

Step	Hyp	Ref	Expression
1		trnij.1	. . . . . . . . . 10 |- F = (x e. RR |-> (x + A))
2	1	trdom 15013	. . . . . . . . 9 |- (A e. RR -> dom F = RR)
3	1	cmpfun 14480	. . . . . . . . 9 |- Fun F
4	2, 3	jctil 316	. . . . . . . 8 |- (A e. RR -> (Fun F /\ dom F = RR))
5		df-fn 4009	. . . . . . . 8 |- (F Fn RR <-> (Fun F /\ dom F = RR))
6	4, 5	sylibr 217	. . . . . . 7 |- (A e. RR -> F Fn RR)
7	1	trran 15014	. . . . . . 7 |- (A e. RR -> ran F = RR)
8		eqimss 2665	. . . . . . . 8 |- (ran F = RR -> ran F C_ RR)
9	8	anim2i 362	. . . . . . 7 |- ((F Fn RR /\ ran F = RR) -> (F Fn RR /\ ran F C_ RR))
10	6, 7, 9	syl11anc 524	. . . . . 6 |- (A e. RR -> (F Fn RR /\ ran F C_ RR))
11		df-f 4010	. . . . . 6 |- (F:RR-->RR <-> (F Fn RR /\ ran F C_ RR))
12	10, 11	sylibr 217	. . . . 5 |- (A e. RR -> F:RR-->RR)
13		readdcl 6455	. . . . . . . . . . 11 |- ((x e. RR /\ A e. RR) -> (x + A) e. RR)
14	13	expcom 403	. . . . . . . . . 10 |- (A e. RR -> (x e. RR -> (x + A) e. RR))
15		readdcl 6455	. . . . . . . . . . 11 |- ((y e. RR /\ A e. RR) -> (y + A) e. RR)
16	15	expcom 403	. . . . . . . . . 10 |- (A e. RR -> (y e. RR -> (y + A) e. RR))
17	14, 16	anim12d 617	. . . . . . . . 9 |- (A e. RR -> ((x e. RR /\ y e. RR) -> ((x + A) e. RR /\ (y + A) e. RR)))
18	17	imp 377	. . . . . . . 8 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> ((x + A) e. RR /\ (y + A) e. RR))
19		simpr 350	. . . . . . . . . . . 12 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> (F` x) = (F` y))
20	1	fveq1i 4682	. . . . . . . . . . . 12 |- (F` x) = ((x e. RR |-> (x + A))` x)
21		df-mpt 5006	. . . . . . . . . . . . . . 15 |- (x e. RR |-> (x + A)) = {<.x, u>. | (x e. RR /\ u = (x + A))}
22		eleq1 1957	. . . . . . . . . . . . . . . . . 18 |- (x = y -> (x e. RR <-> y e. RR))
23	22	adantr 425	. . . . . . . . . . . . . . . . 17 |- ((x = y /\ u = v) -> (x e. RR <-> y e. RR))
24		eqeq1 1890	. . . . . . . . . . . . . . . . . 18 |- (u = v -> (u = (x + A) <-> v = (x + A)))
25		opreq1 4889	. . . . . . . . . . . . . . . . . . 19 |- (x = y -> (x + A) = (y + A))
26	25	eqeq2d 1895	. . . . . . . . . . . . . . . . . 18 |- (x = y -> (v = (x + A) <-> v = (y + A)))
27	24, 26	sylan9bbr 600	. . . . . . . . . . . . . . . . 17 |- ((x = y /\ u = v) -> (u = (x + A) <-> v = (y + A)))
28	23, 27	anbi12d 690	. . . . . . . . . . . . . . . 16 |- ((x = y /\ u = v) -> ((x e. RR /\ u = (x + A)) <-> (y e. RR /\ v = (y + A))))
29	28	cbvopabv 3404	. . . . . . . . . . . . . . 15 |- {<.x, u>. | (x e. RR /\ u = (x + A))} = {<.y, v>. | (y e. RR /\ v = (y + A))}
30	1, 21, 29	3eqtri 1912	. . . . . . . . . . . . . 14 |- F = {<.y, v>. | (y e. RR /\ v = (y + A))}
31		df-mpt 5006	. . . . . . . . . . . . . 14 |- (y e. RR |-> (y + A)) = {<.y, v>. | (y e. RR /\ v = (y + A))}
32	30, 31	eqtr4i 1911	. . . . . . . . . . . . 13 |- F = (y e. RR |-> (y + A))
33	32	fveq1i 4682	. . . . . . . . . . . 12 |- (F` y) = ((y e. RR |-> (y + A))` y)
34	19, 20, 33	3eqtr3g 1952	. . . . . . . . . . 11 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> ((x e. RR |-> (x + A))` x) = ((y e. RR |-> (y + A))` y))
35		simprl 450	. . . . . . . . . . . . 13 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> x e. RR)
36	35	ad2antrr 440	. . . . . . . . . . . 12 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> x e. RR)
37		simplrl 454	. . . . . . . . . . . 12 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> (x + A) e. RR)
38		eqid 1884	. . . . . . . . . . . . 13 |- (x e. RR |-> (x + A)) = (x e. RR |-> (x + A))
39	38	fvopab2b 14476	. . . . . . . . . . . 12 |- ((x e. RR /\ (x + A) e. RR) -> ((x e. RR |-> (x + A))` x) = (x + A))
40	36, 37, 39	syl11anc 524	. . . . . . . . . . 11 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> ((x e. RR |-> (x + A))` x) = (x + A))
41		simplrr 455	. . . . . . . . . . . . . 14 |- (((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) -> y e. RR)
42		simprr 451	. . . . . . . . . . . . . 14 |- (((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) -> (y + A) e. RR)
43	41, 42	jca 310	. . . . . . . . . . . . 13 |- (((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) -> (y e. RR /\ (y + A) e. RR))
44	43	adantr 425	. . . . . . . . . . . 12 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> (y e. RR /\ (y + A) e. RR))
45		eqid 1884	. . . . . . . . . . . . 13 |- (y e. RR |-> (y + A)) = (y e. RR |-> (y + A))
46	45	fvopab2b 14476	. . . . . . . . . . . 12 |- ((y e. RR /\ (y + A) e. RR) -> ((y e. RR |-> (y + A))` y) = (y + A))
47	44, 46	syl 12	. . . . . . . . . . 11 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> ((y e. RR |-> (y + A))` y) = (y + A))
48	34, 40, 47	3eqtr3d 1934	. . . . . . . . . 10 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> (x + A) = (y + A))
49		recn 6466	. . . . . . . . . . . . . 14 |- (x e. RR -> x e. CC)
50	49	ad2antrl 442	. . . . . . . . . . . . 13 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> x e. CC)
51		recn 6466	. . . . . . . . . . . . . 14 |- (y e. RR -> y e. CC)
52	51	ad2antll 443	. . . . . . . . . . . . 13 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> y e. CC)
53		recn 6466	. . . . . . . . . . . . . 14 |- (A e. RR -> A e. CC)
54	53	adantr 425	. . . . . . . . . . . . 13 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> A e. CC)
55	50, 52, 54	3jca 1050	. . . . . . . . . . . 12 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> (x e. CC /\ y e. CC /\ A e. CC))
56	55	ad2antrr 440	. . . . . . . . . . 11 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> (x e. CC /\ y e. CC /\ A e. CC))
57		addcan2 6508	. . . . . . . . . . 11 |- ((x e. CC /\ y e. CC /\ A e. CC) -> ((x + A) = (y + A) <-> x = y))
58	56, 57	syl 12	. . . . . . . . . 10 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> ((x + A) = (y + A) <-> x = y))
59	48, 58	mpbid 212	. . . . . . . . 9 |- ((((A e. RR /\ (x e. RR /\ y e. RR)) /\ ((x + A) e. RR /\ (y + A) e. RR)) /\ (F` x) = (F` y)) -> x = y)
60	59	exp31 407	. . . . . . . 8 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> (((x + A) e. RR /\ (y + A) e. RR) -> ((F` x) = (F` y) -> x = y)))
61	18, 60	mpd 29	. . . . . . 7 |- ((A e. RR /\ (x e. RR /\ y e. RR)) -> ((F` x) = (F` y) -> x = y))
62	61	ex 402	. . . . . 6 |- (A e. RR -> ((x e. RR /\ y e. RR) -> ((F` x) = (F` y) -> x = y)))
63	62	r19.21aivv 2183	. . . . 5 |- (A e. RR -> A.x e. RR A.y e. RR ((F` x) = (F` y) -> x = y))
64	12, 63	jca 310	. . . 4 |- (A e. RR -> (F:RR-->RR /\ A.x e. RR A.y e. RR ((F` x) = (F` y) -> x = y)))
65		dff13 4850	. . . 4 |- (F:RR-1-1->RR <-> (F:RR-->RR /\ A.x e. RR A.y e. RR ((F` x) = (F` y) -> x = y)))
66	64, 65	sylibr 217	. . 3 |- (A e. RR -> F:RR-1-1->RR)
67	6, 7	jca 310	. . . 4 |- (A e. RR -> (F Fn RR /\ ran F = RR))
68		df-fo 4012	. . . 4 |- (F:RR-onto->RR <-> (F Fn RR /\ ran F = RR))
69	67, 68	sylibr 217	. . 3 |- (A e. RR -> F:RR-onto->RR)
70	66, 69	jca 310	. 2 |- (A e. RR -> (F:RR-1-1->RR /\ F:RR-onto->RR))
71		df-f1o 4013	. 2 |- (F:RR-1-1-onto->RR <-> (F:RR-1-1->RR /\ F:RR-onto->RR))
72	70, 71	sylibr 217	1 |- (A e. RR -> F:RR-1-1-onto->RR)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  {copab 3395  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  CCcc 6384  RRcr 6385   + caddc 6389

This theorem is referenced by:  cnvtr 15016

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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  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-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-sub 6511  df-neg 6513

Copyright terms: Public domain