Theorem isoresbr 5831

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 5830	. . . 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 5558	. . . . . 6 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
3		fvres 5558	. . . . . 6 |- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) )
4	2, 3	breqan12d 4034	. . . . 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 2572	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 2160   A.wral 2468   class class class wbr 4018   |` cres 4646   "cima 4647   ` cfv 5235   Isom wiso 5236

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-res 4656  df-iota 5196  df-fv 5243  df-isom 5244

This theorem is referenced by: (None)