Theorem ismtycnv 15949

Description: The inverse of an isometry is an isometry.

Assertion

Ref	Expression
ismtycnv	|- ((M e. Met /\ N e. Met) -> (F e. (MIsmtyN) -> `'F e. (NIsmtyM)))

Proof of Theorem ismtycnv

Step	Hyp	Ref	Expression
1		f1ocnv 4651	. . . . 5 |- (F:dom dom M-1-1-onto->dom dom N -> `'F:dom dom N-1-1-onto->dom dom M)
2	1	adantr 425	. . . 4 |- ((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:dom dom N-1-1-onto->dom dom M)
3		f1ocnvdm 4860	. . . . . . . . . . . 12 |- ((F:dom dom M-1-1-onto->dom dom N /\ u e. dom dom N) -> (`'F` u) e. dom dom M)
4	3	ex 402	. . . . . . . . . . 11 |- (F:dom dom M-1-1-onto->dom dom N -> (u e. dom dom N -> (`'F` u) e. dom dom M))
5		f1ocnvdm 4860	. . . . . . . . . . . 12 |- ((F:dom dom M-1-1-onto->dom dom N /\ v e. dom dom N) -> (`'F` v) e. dom dom M)
6	5	ex 402	. . . . . . . . . . 11 |- (F:dom dom M-1-1-onto->dom dom N -> (v e. dom dom N -> (`'F` v) e. dom dom M))
7	4, 6	anim12d 617	. . . . . . . . . 10 |- (F:dom dom M-1-1-onto->dom dom N -> ((u e. dom dom N /\ v e. dom dom N) -> ((`'F` u) e. dom dom M /\ (`'F` v) e. dom dom M)))
8	7	adantr 425	. . . . . . . . 9 |- ((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))) -> ((u e. dom dom N /\ v e. dom dom N) -> ((`'F` u) e. dom dom M /\ (`'F` v) e. dom dom M)))
9	8	imdistani 491	. . . . . . . 8 |- (((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))) /\ (u e. dom dom N /\ v e. dom dom N)) -> ((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` u) e. dom dom M /\ (`'F` v) e. dom dom M)))
10		opreq1 4889	. . . . . . . . . . . 12 |- (x = (`'F` u) -> (xMy) = ((`'F` u)My))
11		fveq2 4681	. . . . . . . . . . . . 13 |- (x = (`'F` u) -> (F` x) = (F` (`'F` u)))
12	11	opreq1d 4897	. . . . . . . . . . . 12 |- (x = (`'F` u) -> ((F` x)N(F` y)) = ((F` (`'F` u))N(F` y)))
13	10, 12	eqeq12d 1899	. . . . . . . . . . 11 |- (x = (`'F` u) -> ((xMy) = ((F` x)N(F` y)) <-> ((`'F` u)My) = ((F` (`'F` u))N(F` y))))
14		opreq2 4890	. . . . . . . . . . . 12 |- (y = (`'F` v) -> ((`'F` u)My) = ((`'F` u)M(`'F` v)))
15		fveq2 4681	. . . . . . . . . . . . 13 |- (y = (`'F` v) -> (F` y) = (F` (`'F` v)))
16	15	opreq2d 4898	. . . . . . . . . . . 12 |- (y = (`'F` v) -> ((F` (`'F` u))N(F` y)) = ((F` (`'F` u))N(F` (`'F` v))))
17	14, 16	eqeq12d 1899	. . . . . . . . . . 11 |- (y = (`'F` v) -> (((`'F` u)My) = ((F` (`'F` u))N(F` y)) <-> ((`'F` u)M(`'F` v)) = ((F` (`'F` u))N(F` (`'F` v)))))
18	13, 17	rcla42v 2384	. . . . . . . . . 10 |- (((`'F` u) e. dom dom M /\ (`'F` v) e. dom dom M) -> (A.x e. dom dom MA.y e. dom dom M(xMy) = ((F` x)N(F` y)) -> ((`'F` u)M(`'F` v)) = ((F` (`'F` u))N(F` (`'F` v)))))
19	18	impcom 378	. . . . . . . . 9 |- ((A.x e. dom dom MA.y e. dom dom M(xMy) = ((F` x)N(F` y)) /\ ((`'F` u) e. dom dom M /\ (`'F` v) e. dom dom M)) -> ((`'F` u)M(`'F` v)) = ((F` (`'F` u))N(F` (`'F` v))))
20	19	adantll 428	. . . . . . . 8 |- (((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` u) e. dom dom M /\ (`'F` v) e. dom dom M)) -> ((`'F` u)M(`'F` v)) = ((F` (`'F` u))N(F` (`'F` v))))
21	9, 20	syl 12	. . . . . . 7 |- (((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))) /\ (u e. dom dom N /\ v e. dom dom N)) -> ((`'F` u)M(`'F` v)) = ((F` (`'F` u))N(F` (`'F` v))))
22		f1ocnvfv2 4855	. . . . . . . . . 10 |- ((F:dom dom M-1-1-onto->dom dom N /\ u e. dom dom N) -> (F` (`'F` u)) = u)
23	22	adantrr 431	. . . . . . . . 9 |- ((F:dom dom M-1-1-onto->dom dom N /\ (u e. dom dom N /\ v e. dom dom N)) -> (F` (`'F` u)) = u)
24		f1ocnvfv2 4855	. . . . . . . . . 10 |- ((F:dom dom M-1-1-onto->dom dom N /\ v e. dom dom N) -> (F` (`'F` v)) = v)
25	24	adantrl 430	. . . . . . . . 9 |- ((F:dom dom M-1-1-onto->dom dom N /\ (u e. dom dom N /\ v e. dom dom N)) -> (F` (`'F` v)) = v)
26	23, 25	opreq12d 4900	. . . . . . . 8 |- ((F:dom dom M-1-1-onto->dom dom N /\ (u e. dom dom N /\ v e. dom dom N)) -> ((F` (`'F` u))N(F` (`'F` v))) = (uNv))
27	26	adantlr 429	. . . . . . 7 |- (((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))) /\ (u e. dom dom N /\ v e. dom dom N)) -> ((F` (`'F` u))N(F` (`'F` v))) = (uNv))
28	21, 27	eqtr2d 1926	. . . . . 6 |- (((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))) /\ (u e. dom dom N /\ v e. dom dom N)) -> (uNv) = ((`'F` u)M(`'F` v)))
29	28	ex 402	. . . . 5 |- ((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))) -> ((u e. dom dom N /\ v e. dom dom N) -> (uNv) = ((`'F` u)M(`'F` v))))
30	29	r19.21aivv 2183	. . . 4 |- ((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))) -> A.u e. dom dom NA.v e. dom dom N(uNv) = ((`'F` u)M(`'F` v)))
31	2, 30	jca 310	. . 3 |- ((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:dom dom N-1-1-onto->dom dom M /\ A.u e. dom dom NA.v e. dom dom N(uNv) = ((`'F` u)M(`'F` v))))
32	31	a1i 8	. 2 |- ((M e. Met /\ N e. Met) -> ((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:dom dom N-1-1-onto->dom dom M /\ A.u e. dom dom NA.v e. dom dom N(uNv) = ((`'F` u)M(`'F` v)))))
33		eqid 1884	. . 3 |- dom dom M = dom dom M
34		eqid 1884	. . 3 |- dom dom N = dom dom N
35	33, 34	isismty 15948	. 2 |- ((M e. Met /\ N e. Met) -> (F e. (MIsmtyN) <-> (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)))))
36	34, 33	isismty 15948	. . 3 |- ((N e. Met /\ M e. Met) -> (`'F e. (NIsmtyM) <-> (`'F:dom dom N-1-1-onto->dom dom M /\ A.u e. dom dom NA.v e. dom dom N(uNv) = ((`'F` u)M(`'F` v)))))
37	36	ancoms 484	. 2 |- ((M e. Met /\ N e. Met) -> (`'F e. (NIsmtyM) <-> (`'F:dom dom N-1-1-onto->dom dom M /\ A.u e. dom dom NA.v e. dom dom N(uNv) = ((`'F` u)M(`'F` v)))))
38	32, 35, 37	3imtr4d 602	1 |- ((M e. Met /\ N e. Met) -> (F e. (MIsmtyN) -> `'F e. (NIsmtyM)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  `'ccnv 3985  dom cdm 3986  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Metcme 9066  Ismtycismty 15945

This theorem is referenced by:  ismtybnd 15953

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-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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