Theorem isorel 4871

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 3341	. . . . 5 |- (x = C -> (xRy <-> CRy))
2		fveq2 4681	. . . . . 6 |- (x = C -> (H` x) = (H` C))
3	2	breq1d 3348	. . . . 5 |- (x = C -> ((H` x)S(H` y) <-> (H` C)S(H` y)))
4	1, 3	bibi12d 691	. . . 4 |- (x = C -> ((xRy <-> (H` x)S(H` y)) <-> (CRy <-> (H` C)S(H` y))))
5		breq2 3342	. . . . 5 |- (y = D -> (CRy <-> CRD))
6		fveq2 4681	. . . . . 6 |- (y = D -> (H` y) = (H` D))
7	6	breq2d 3350	. . . . 5 |- (y = D -> ((H` C)S(H` y) <-> (H` C)S(H` D)))
8	5, 7	bibi12d 691	. . . 4 |- (y = D -> ((CRy <-> (H` C)S(H` y)) <-> (CRD <-> (H` C)S(H` D))))
9	4, 8	rcla42v 2384	. . 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 4015	. . . 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	simprbi 353	. . 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 63	. 2 |- (H Isom R, S (A, B) -> ((C e. A /\ D e. A) -> (CRD <-> (H` C)S(H` D))))
13	12	imp 377	1 |- ((H Isom R, S (A, B) /\ (C e. A /\ D e. A)) -> (CRD <-> (H` C)S(H` D)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  -1-1-onto->wf1o 3997  ` cfv 3998   Isom wiso 3999

This theorem is referenced by:  isomin 4876  isoini 4877  isowe 4880  ordiso 5683  ordisoOLD 15374  smoiso 16453

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-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

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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-iso 4015

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