Theorem isoresbr 5960

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 5959	. . . 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 5672	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
3		fvres 5672	. . . . . 6 |- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) )
4	2, 3	breqan12d 4109	. . . . 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 2615	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 2202   A.wral 2511   class class class wbr 4093   |` cres 4733   "cima 4734   ` cfv 5333   Isom wiso 5334

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-res 4743  df-iota 5293  df-fv 5341  df-isom 5342

This theorem is referenced by: (None)