Theorem isorel 3885

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 2617	. . . . 5 |- (x = C -> (xRy <-> CRy))
2		fveq2 3715	. . . . . 6 |- (x = C -> (H` x) = (H` C))
3	2	breq1d 2624	. . . . 5 |- (x = C -> ((H` x)S(H` y) <-> (H` C)S(H` y)))
4	1, 3	bibi12d 628	. . . 4 |- (x = C -> ((xRy <-> (H` x)S(H` y)) <-> (CRy <-> (H` C)S(H` y))))
5		breq2 2618	. . . . 5 |- (y = D -> (CRy <-> CRD))
6		fveq2 3715	. . . . . 6 |- (y = D -> (H` y) = (H` D))
7	6	breq2d 2625	. . . . 5 |- (y = D -> ((H` C)S(H` y) <-> (H` C)S(H` D)))
8	5, 7	bibi12d 628	. . . 4 |- (y = D -> ((CRy <-> (H` C)S(H` y)) <-> (CRD <-> (H` C)S(H` D))))
9	4, 8	rcla42v 1876	. . 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 3194	. . . 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 954   e. wcel 956  A.wral 1642   class class class wbr 2614  -1-1-onto->wf1o 3176  ` cfv 3177   Isom wiso 3178

This theorem is referenced by:  isomin 3890  isoini 3891  isowe 3894

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-iso 3194

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