Theorem isorel 3894

Description: An isomorphism connects binary relations via its function values.

Assertion

Ref	Expression
isorel	|- ((H Isom R, S (A, B) /\ (C e. A /\ D e. A)) -> (CRD <-> (H` C)S(H` D)))

Proof of Theorem isorel

Step	Hyp	Ref	Expression
1		breq1 2622	. . . . 5 |- (x = C -> (xRy <-> CRy))
2		fveq2 3724	. . . . . 6 |- (x = C -> (H` x) = (H` C))
3	2	breq1d 2629	. . . . 5 |- (x = C -> ((H` x)S(H` y) <-> (H` C)S(H` y)))
4	1, 3	bibi12d 629	. . . 4 |- (x = C -> ((xRy <-> (H` x)S(H` y)) <-> (CRy <-> (H` C)S(H` y))))
5		breq2 2623	. . . . 5 |- (y = D -> (CRy <-> CRD))
6		fveq2 3724	. . . . . 6 |- (y = D -> (H` y) = (H` D))
7	6	breq2d 2630	. . . . 5 |- (y = D -> ((H` C)S(H` y) <-> (H` C)S(H` D)))
8	5, 7	bibi12d 629	. . . 4 |- (y = D -> ((CRy <-> (H` C)S(H` y)) <-> (CRD <-> (H` C)S(H` D))))
9	4, 8	rcla42v 1880	. . 3 |- ((C e. A /\ D e. A) -> (A.x e. A A.y e. A (xRy <-> (H` x)S(H` y)) -> (CRD <-> (H` C)S(H` D))))
10		df-iso 3199	. . . 4 |- (H Isom R, S (A, B) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))))
11	10	pm3.27bi 326	. . 3 |- (H Isom R, S (A, B) -> A.x e. A A.y e. A (xRy <-> (H` x)S(H` y)))
12	9, 11	syl5com 52	. 2 |- (H Isom R, S (A, B) -> ((C e. A /\ D e. A) -> (CRD <-> (H` C)S(H` D))))
13	12	imp 350	1 |- ((H Isom R, S (A, B) /\ (C e. A /\ D e. A)) -> (CRD <-> (H` C)S(H` D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   class class class wbr 2619  -1-1-onto->wf1o 3181  ` cfv 3182   Isom wiso 3183

This theorem is referenced by:  isomin 3899  isoini 3900  isowe 3903

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-iso 3199

Copyright terms: Public domain