Theorem isoresbr 5813

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 5812	. . . 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 5541	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
3		fvres 5541	. . . . . 6 |- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) )
4	2, 3	breqan12d 4021	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( ( F |` A ) ` x ) S ( ( F |` A ) ` y ) <-> ( F ` x ) S ( F ` y ) ) )
5	4	adantl 277	. . . 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 188	. . 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 144	. 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 2559	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 104   <-> wb 105   e. wcel 2148   A.wral 2455   class class class wbr 4005   |` cres 4630   "cima 4631   ` cfv 5218   Isom wiso 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-res 4640  df-iota 5180  df-fv 5226  df-isom 5227

This theorem is referenced by: (None)