Theorem f1oiso 4881

Description: Any one-to-one onto function determines an isomorphism with an induced relation S. Proposition 6.33 of [TakeutiZaring] p. 34.

Assertion

Ref	Expression
f1oiso	|- ((H:A-1-1-onto->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> H Isom R, S (A, B))

Distinct variable groups:   x,y,z,w,A   x,B,y   x,H,y,z,w   x,R,y,z,w

Proof of Theorem f1oiso

Step	Hyp	Ref	Expression
1		df-iso 4015	. 2 |- (H Isom R, S (A, B) <-> (H:A-1-1-onto->B /\ A.v e. A A.u e. A (vRu <-> (H` v)S(H` u))))
2		simpl 346	. 2 |- ((H:A-1-1-onto->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> H:A-1-1-onto->B)
3		eleq2 1958	. . . . . . . . 9 |- (S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)} -> (<.(H` v), (H` u)>. e. S <-> <.(H` v), (H` u)>. e. {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}))
4		f1fveq 4852	. . . . . . . . . . . . . . . . . . . 20 |- ((H:A-1-1->B /\ (v e. A /\ x e. A)) -> ((H` v) = (H` x) <-> v = x))
5		eqcom 1886	. . . . . . . . . . . . . . . . . . . 20 |- (v = x <-> x = v)
6	4, 5	syl6bb 595	. . . . . . . . . . . . . . . . . . 19 |- ((H:A-1-1->B /\ (v e. A /\ x e. A)) -> ((H` v) = (H` x) <-> x = v))
7	6	anassrs 489	. . . . . . . . . . . . . . . . . 18 |- (((H:A-1-1->B /\ v e. A) /\ x e. A) -> ((H` v) = (H` x) <-> x = v))
8	7	anbi1d 679	. . . . . . . . . . . . . . . . 17 |- (((H:A-1-1->B /\ v e. A) /\ x e. A) -> (((H` v) = (H` x) /\ ((H` u) = (H` y) /\ xRy)) <-> (x = v /\ ((H` u) = (H` y) /\ xRy))))
9		anass 487	. . . . . . . . . . . . . . . . 17 |- ((((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> ((H` v) = (H` x) /\ ((H` u) = (H` y) /\ xRy)))
10	8, 9	syl5bb 591	. . . . . . . . . . . . . . . 16 |- (((H:A-1-1->B /\ v e. A) /\ x e. A) -> ((((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> (x = v /\ ((H` u) = (H` y) /\ xRy))))
11	10	rexbidv 2124	. . . . . . . . . . . . . . 15 |- (((H:A-1-1->B /\ v e. A) /\ x e. A) -> (E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> E.y e. A (x = v /\ ((H` u) = (H` y) /\ xRy))))
12		r19.42v 2237	. . . . . . . . . . . . . . 15 |- (E.y e. A (x = v /\ ((H` u) = (H` y) /\ xRy)) <-> (x = v /\ E.y e. A ((H` u) = (H` y) /\ xRy)))
13	11, 12	syl6bb 595	. . . . . . . . . . . . . 14 |- (((H:A-1-1->B /\ v e. A) /\ x e. A) -> (E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> (x = v /\ E.y e. A ((H` u) = (H` y) /\ xRy))))
14	13	rexbidva 2120	. . . . . . . . . . . . 13 |- ((H:A-1-1->B /\ v e. A) -> (E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> E.x e. A (x = v /\ E.y e. A ((H` u) = (H` y) /\ xRy))))
15		breq1 3341	. . . . . . . . . . . . . . . . 17 |- (x = v -> (xRy <-> vRy))
16	15	anbi2d 678	. . . . . . . . . . . . . . . 16 |- (x = v -> (((H` u) = (H` y) /\ xRy) <-> ((H` u) = (H` y) /\ vRy)))
17	16	rexbidv 2124	. . . . . . . . . . . . . . 15 |- (x = v -> (E.y e. A ((H` u) = (H` y) /\ xRy) <-> E.y e. A ((H` u) = (H` y) /\ vRy)))
18	17	ceqsrexv 2394	. . . . . . . . . . . . . 14 |- (v e. A -> (E.x e. A (x = v /\ E.y e. A ((H` u) = (H` y) /\ xRy)) <-> E.y e. A ((H` u) = (H` y) /\ vRy)))
19	18	adantl 424	. . . . . . . . . . . . 13 |- ((H:A-1-1->B /\ v e. A) -> (E.x e. A (x = v /\ E.y e. A ((H` u) = (H` y) /\ xRy)) <-> E.y e. A ((H` u) = (H` y) /\ vRy)))
20	14, 19	bitrd 587	. . . . . . . . . . . 12 |- ((H:A-1-1->B /\ v e. A) -> (E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> E.y e. A ((H` u) = (H` y) /\ vRy)))
21		f1fveq 4852	. . . . . . . . . . . . . . . . 17 |- ((H:A-1-1->B /\ (u e. A /\ y e. A)) -> ((H` u) = (H` y) <-> u = y))
22		eqcom 1886	. . . . . . . . . . . . . . . . 17 |- (u = y <-> y = u)
23	21, 22	syl6bb 595	. . . . . . . . . . . . . . . 16 |- ((H:A-1-1->B /\ (u e. A /\ y e. A)) -> ((H` u) = (H` y) <-> y = u))
24	23	anassrs 489	. . . . . . . . . . . . . . 15 |- (((H:A-1-1->B /\ u e. A) /\ y e. A) -> ((H` u) = (H` y) <-> y = u))
25	24	anbi1d 679	. . . . . . . . . . . . . 14 |- (((H:A-1-1->B /\ u e. A) /\ y e. A) -> (((H` u) = (H` y) /\ vRy) <-> (y = u /\ vRy)))
26	25	rexbidva 2120	. . . . . . . . . . . . 13 |- ((H:A-1-1->B /\ u e. A) -> (E.y e. A ((H` u) = (H` y) /\ vRy) <-> E.y e. A (y = u /\ vRy)))
27		breq2 3342	. . . . . . . . . . . . . . 15 |- (y = u -> (vRy <-> vRu))
28	27	ceqsrexv 2394	. . . . . . . . . . . . . 14 |- (u e. A -> (E.y e. A (y = u /\ vRy) <-> vRu))
29	28	adantl 424	. . . . . . . . . . . . 13 |- ((H:A-1-1->B /\ u e. A) -> (E.y e. A (y = u /\ vRy) <-> vRu))
30	26, 29	bitrd 587	. . . . . . . . . . . 12 |- ((H:A-1-1->B /\ u e. A) -> (E.y e. A ((H` u) = (H` y) /\ vRy) <-> vRu))
31	20, 30	sylan9bb 599	. . . . . . . . . . 11 |- (((H:A-1-1->B /\ v e. A) /\ (H:A-1-1->B /\ u e. A)) -> (E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> vRu))
32	31	anandis 570	. . . . . . . . . 10 |- ((H:A-1-1->B /\ (v e. A /\ u e. A)) -> (E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy) <-> vRu))
33		fvex 4689	. . . . . . . . . . 11 |- (H` v) e. _V
34		fvex 4689	. . . . . . . . . . 11 |- (H` u) e. _V
35		eqeq1 1890	. . . . . . . . . . . . . 14 |- (z = (H` v) -> (z = (H` x) <-> (H` v) = (H` x)))
36	35	anbi1d 679	. . . . . . . . . . . . 13 |- (z = (H` v) -> ((z = (H` x) /\ w = (H` y)) <-> ((H` v) = (H` x) /\ w = (H` y))))
37	36	anbi1d 679	. . . . . . . . . . . 12 |- (z = (H` v) -> (((z = (H` x) /\ w = (H` y)) /\ xRy) <-> (((H` v) = (H` x) /\ w = (H` y)) /\ xRy)))
38	37	2rexbidv 2141	. . . . . . . . . . 11 |- (z = (H` v) -> (E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy) <-> E.x e. A E.y e. A (((H` v) = (H` x) /\ w = (H` y)) /\ xRy)))
39		eqeq1 1890	. . . . . . . . . . . . . 14 |- (w = (H` u) -> (w = (H` y) <-> (H` u) = (H` y)))
40	39	anbi2d 678	. . . . . . . . . . . . 13 |- (w = (H` u) -> (((H` v) = (H` x) /\ w = (H` y)) <-> ((H` v) = (H` x) /\ (H` u) = (H` y))))
41	40	anbi1d 679	. . . . . . . . . . . 12 |- (w = (H` u) -> ((((H` v) = (H` x) /\ w = (H` y)) /\ xRy) <-> (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy)))
42	41	2rexbidv 2141	. . . . . . . . . . 11 |- (w = (H` u) -> (E.x e. A E.y e. A (((H` v) = (H` x) /\ w = (H` y)) /\ xRy) <-> E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy)))
43	33, 34, 38, 42	opelopab 3570	. . . . . . . . . 10 |- (<.(H` v), (H` u)>. e. {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)} <-> E.x e. A E.y e. A (((H` v) = (H` x) /\ (H` u) = (H` y)) /\ xRy))
44	32, 43	syl5bb 591	. . . . . . . . 9 |- ((H:A-1-1->B /\ (v e. A /\ u e. A)) -> (<.(H` v), (H` u)>. e. {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)} <-> vRu))
45	3, 44	sylan9bbr 600	. . . . . . . 8 |- (((H:A-1-1->B /\ (v e. A /\ u e. A)) /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> (<.(H` v), (H` u)>. e. S <-> vRu))
46	45	an1rs 547	. . . . . . 7 |- (((H:A-1-1->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) /\ (v e. A /\ u e. A)) -> (<.(H` v), (H` u)>. e. S <-> vRu))
47		df-br 3339	. . . . . . 7 |- ((H` v)S(H` u) <-> <.(H` v), (H` u)>. e. S)
48	46, 47	syl5rbb 592	. . . . . 6 |- (((H:A-1-1->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) /\ (v e. A /\ u e. A)) -> (vRu <-> (H` v)S(H` u)))
49	48	exp32 408	. . . . 5 |- ((H:A-1-1->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> (v e. A -> (u e. A -> (vRu <-> (H` v)S(H` u)))))
50	49	r19.21adv 2181	. . . 4 |- ((H:A-1-1->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> (v e. A -> A.u e. A (vRu <-> (H` v)S(H` u))))
51	50	r19.21aiv 2175	. . 3 |- ((H:A-1-1->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> A.v e. A A.u e. A (vRu <-> (H` v)S(H` u)))
52		f1of1 4634	. . 3 |- (H:A-1-1-onto->B -> H:A-1-1->B)
53	51, 52	sylan 497	. 2 |- ((H:A-1-1-onto->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> A.v e. A A.u e. A (vRu <-> (H` v)S(H` u)))
54	1, 2, 53	sylanbrc 527	1 |- ((H:A-1-1-onto->B /\ S = {<.z, w>. | E.x e. A E.y e. A ((z = (H` x) /\ w = (H` y)) /\ xRy)}) -> H Isom R, S (A, B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  <.cop 3046   class class class wbr 3338  {copab 3395  -1-1->wf1 3995  -1-1-onto->wf1o 3997  ` cfv 3998   Isom wiso 3999

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-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-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-f1o 4013  df-fv 4014  df-iso 4015

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