Theorem resvval2 29829

Description: Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)

Hypotheses

Ref	Expression
resvsca.r	|- R = ( W ↾v A )
resvsca.f	|- F = (Scalar ` W )
resvsca.b	|- B = ( Base ` F )

Assertion

Ref	Expression
resvval2	|- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> R = ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) )

Proof of Theorem resvval2

Step	Hyp	Ref	Expression
1		resvsca.r	. . . 4 |- R = ( W ↾v A )
2		resvsca.f	. . . 4 |- F = (Scalar ` W )
3		resvsca.b	. . . 4 |- B = ( Base ` F )
4	1, 2, 3	resvval 29827	. . 3 |- ( ( W e. X /\ A e. Y ) -> R = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )
5		iffalse 4095	. . 3 |- ( -. B C_ A -> if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) = ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) )
6	4, 5	sylan9eqr 2678	. 2 |- ( ( -. B C_ A /\ ( W e. X /\ A e. Y ) ) -> R = ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) )
7	6	3impb 1260	1 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> R = ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ifcif 4086   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   ↾_v cresv 29824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-resv 29825

This theorem is referenced by:  resvsca  29830  resvlem  29831