Theorem ismtyval 15947

Description: The set of isometries between two metric spaces.

Hypotheses

Ref	Expression
ismtyval.1	|- X = dom dom M
ismtyval.2	|- Y = dom dom N

Assertion

Ref	Expression
ismtyval	|- ((M e. Met /\ N e. Met) -> (MIsmtyN) = {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))})

Distinct variable groups:   f,M,x,y   f,N,x,y   f,X,x,y   f,Y,x,y

Proof of Theorem ismtyval

Step	Hyp	Ref	Expression
1		ssexg 3457	. . 3 |- (({f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))} C_ {f | f:X-->Y} /\ {f | f:X-->Y} e. _V) -> {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))} e. _V)
2		f1of 4635	. . . . 5 |- (f:X-1-1-onto->Y -> f:X-->Y)
3	2	adantr 425	. . . 4 |- ((f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y))) -> f:X-->Y)
4	3	ss2abi 2679	. . 3 |- {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))} C_ {f | f:X-->Y}
5		mapex 5387	. . . 4 |- ((X e. _V /\ Y e. _V) -> {f | f:X-->Y} e. _V)
6		dmexg 4206	. . . . . 6 |- (M e. Met -> dom M e. _V)
7		dmexg 4206	. . . . . 6 |- (dom M e. _V -> dom dom M e. _V)
8	6, 7	syl 12	. . . . 5 |- (M e. Met -> dom dom M e. _V)
9		ismtyval.1	. . . . 5 |- X = dom dom M
10	8, 9	syl5eqel 1975	. . . 4 |- (M e. Met -> X e. _V)
11		dmexg 4206	. . . . . 6 |- (N e. Met -> dom N e. _V)
12		dmexg 4206	. . . . . 6 |- (dom N e. _V -> dom dom N e. _V)
13	11, 12	syl 12	. . . . 5 |- (N e. Met -> dom dom N e. _V)
14		ismtyval.2	. . . . 5 |- Y = dom dom N
15	13, 14	syl5eqel 1975	. . . 4 |- (N e. Met -> Y e. _V)
16	5, 10, 15	syl2an 503	. . 3 |- ((M e. Met /\ N e. Met) -> {f | f:X-->Y} e. _V)
17	1, 4, 16	sylancr 526	. 2 |- ((M e. Met /\ N e. Met) -> {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))} e. _V)
18		dmeq 4157	. . . . . . . 8 |- (m = M -> dom m = dom M)
19	18	dmeqd 4159	. . . . . . 7 |- (m = M -> dom dom m = dom dom M)
20	19, 9	syl6eqr 1946	. . . . . 6 |- (m = M -> dom dom m = X)
21		f1oeq2 4631	. . . . . 6 |- (dom dom m = X -> (f:dom dom m-1-1-onto->dom dom n <-> f:X-1-1-onto->dom dom n))
22	20, 21	syl 12	. . . . 5 |- (m = M -> (f:dom dom m-1-1-onto->dom dom n <-> f:X-1-1-onto->dom dom n))
23		opreq 4888	. . . . . . . 8 |- (m = M -> (xmy) = (xMy))
24	23	eqeq1d 1892	. . . . . . 7 |- (m = M -> ((xmy) = ((f` x)n(f` y)) <-> (xMy) = ((f` x)n(f` y))))
25	20, 24	raleqbidv 2274	. . . . . 6 |- (m = M -> (A.y e. dom dom m(xmy) = ((f` x)n(f` y)) <-> A.y e. X (xMy) = ((f` x)n(f` y))))
26	20, 25	raleqbidv 2274	. . . . 5 |- (m = M -> (A.x e. dom dom mA.y e. dom dom m(xmy) = ((f` x)n(f` y)) <-> A.x e. X A.y e. X (xMy) = ((f` x)n(f` y))))
27	22, 26	anbi12d 690	. . . 4 |- (m = M -> ((f:dom dom m-1-1-onto->dom dom n /\ A.x e. dom dom mA.y e. dom dom m(xmy) = ((f` x)n(f` y))) <-> (f:X-1-1-onto->dom dom n /\ A.x e. X A.y e. X (xMy) = ((f` x)n(f` y)))))
28	27	abbidv 2008	. . 3 |- (m = M -> {f | (f:dom dom m-1-1-onto->dom dom n /\ A.x e. dom dom mA.y e. dom dom m(xmy) = ((f` x)n(f` y)))} = {f | (f:X-1-1-onto->dom dom n /\ A.x e. X A.y e. X (xMy) = ((f` x)n(f` y)))})
29		dmeq 4157	. . . . . . . 8 |- (n = N -> dom n = dom N)
30	29	dmeqd 4159	. . . . . . 7 |- (n = N -> dom dom n = dom dom N)
31	30, 14	syl6eqr 1946	. . . . . 6 |- (n = N -> dom dom n = Y)
32		f1oeq3 4632	. . . . . 6 |- (dom dom n = Y -> (f:X-1-1-onto->dom dom n <-> f:X-1-1-onto->Y))
33	31, 32	syl 12	. . . . 5 |- (n = N -> (f:X-1-1-onto->dom dom n <-> f:X-1-1-onto->Y))
34		opreq 4888	. . . . . . 7 |- (n = N -> ((f` x)n(f` y)) = ((f` x)N(f` y)))
35	34	eqeq2d 1895	. . . . . 6 |- (n = N -> ((xMy) = ((f` x)n(f` y)) <-> (xMy) = ((f` x)N(f` y))))
36	35	2ralbidv 2140	. . . . 5 |- (n = N -> (A.x e. X A.y e. X (xMy) = ((f` x)n(f` y)) <-> A.x e. X A.y e. X (xMy) = ((f` x)N(f` y))))
37	33, 36	anbi12d 690	. . . 4 |- (n = N -> ((f:X-1-1-onto->dom dom n /\ A.x e. X A.y e. X (xMy) = ((f` x)n(f` y))) <-> (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))))
38	37	abbidv 2008	. . 3 |- (n = N -> {f | (f:X-1-1-onto->dom dom n /\ A.x e. X A.y e. X (xMy) = ((f` x)n(f` y)))} = {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))})
39		df-ismty 15946	. . 3 |- Ismty = {<.<.m, n>., z>. | ((m e. Met /\ n e. Met) /\ z = {f | (f:dom dom m-1-1-onto->dom dom n /\ A.x e. dom dom mA.y e. dom dom m(xmy) = ((f` x)n(f` y)))})}
40	28, 38, 39	oprabval2g 4956	. 2 |- ((M e. Met /\ N e. Met /\ {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))} e. _V) -> (MIsmtyN) = {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))})
41	17, 40	mpd3an3 1192	1 |- ((M e. Met /\ N e. Met) -> (MIsmtyN) = {f | (f:X-1-1-onto->Y /\ A.x e. X A.y e. X (xMy) = ((f` x)N(f` y)))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593  dom cdm 3986  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Metcme 9066  Ismtycismty 15945

This theorem is referenced by:  isismty 15948

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-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-sbc 2454  df-csb 2541  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-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-ismty 15946

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