Theorem isoresbr 5662

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 5661	. . . 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 5397	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
3		fvres 5397	. . . . . 6 |- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) )
4	2, 3	breqan12d 3908	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( ( F |` A ) ` x ) S ( ( F |` A ) ` y ) <-> ( F ` x ) S ( F ` y ) ) )
5	4	adantl 273	. . . 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 187	. . 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 143	. 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 2486	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 103   <-> wb 104   e. wcel 1461   A.wral 2388   class class class wbr 3893   |` cres 4499   "cima 4500   ` cfv 5079   Isom wiso 5080

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-xp 4503  df-res 4509  df-iota 5044  df-fv 5087  df-isom 5088

This theorem is referenced by: (None)