Theorem isoresbr 5449

Description: A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)

Assertion

Ref	Expression
isoresbr	|- ( ( F |` A ) Isom R , S ( A , ( F " A ) ) -> A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) )

Distinct variable groups:   x, y, A   x, F, y   x, R, y   x, S, y

Proof of Theorem isoresbr

Step	Hyp	Ref	Expression
1		isorel 5448	. . . 4 |- ( ( ( F |` A ) Isom R , S ( A , ( F " A ) ) /\ ( x e. A /\ y e. A ) ) -> ( x R y <-> ( ( F |` A ) ` x ) S ( ( F |` A ) ` y ) ) )
2		fvres 5198	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
3		fvres 5198	. . . . . 6 |- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) )
4	2, 3	breqan12d 3779	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( ( F |` A ) ` x ) S ( ( F |` A ) ` y ) <-> ( F ` x ) S ( F ` y ) ) )
5	4	adantl 262	. . . 4 |- ( ( ( F |` A ) Isom R , S ( A , ( F " A ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( ( F |` A ) ` x ) S ( ( F |` A ) ` y ) <-> ( F ` x ) S ( F ` y ) ) )
6	1, 5	bitrd 177	. . 3 |- ( ( ( F |` A ) Isom R , S ( A , ( F " A ) ) /\ ( x e. A /\ y e. A ) ) -> ( x R y <-> ( F ` x ) S ( F ` y ) ) )
7	6	biimpd 132	. 2 |- ( ( ( F |` A ) Isom R , S ( A , ( F " A ) ) /\ ( x e. A /\ y e. A ) ) -> ( x R y -> ( F ` x ) S ( F ` y ) ) )
8	7	ralrimivva 2401	1 |- ( ( F |` A ) Isom R , S ( A , ( F " A ) ) -> A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   A.wral 2306   class class class wbr 3764   |` cres 4347   "cima 4348   ` cfv 4902   Isom wiso 4903

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-isom 4911

This theorem is referenced by: (None)