Theorem pjhval 28256

Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
pjhval	|- ( ( H e. CH /\ A e. ~H ) -> ( ( proj h ` H ) ` A ) = ( iota_ x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) )

Distinct variable groups:   x, y, H   x, A, y

Proof of Theorem pjhval

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjhfval 28255	. . 3 |- ( H e. CH -> ( proj h ` H ) = ( z e. ~H |-> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) ) )
2	1	fveq1d 6193	. 2 |- ( H e. CH -> ( ( proj h ` H ) ` A ) = ( ( z e. ~H |-> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) ) ` A ) )
3		eqeq1 2626	. . . . 5 |- ( z = A -> ( z = ( x +h y ) <-> A = ( x +h y ) ) )
4	3	rexbidv 3052	. . . 4 |- ( z = A -> ( E. y e. ( _|_ ` H ) z = ( x +h y ) <-> E. y e. ( _|_ ` H ) A = ( x +h y ) ) )
5	4	riotabidv 6613	. . 3 |- ( z = A -> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) = ( iota_ x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) )
6		eqid 2622	. . 3 |- ( z e. ~H |-> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) ) = ( z e. ~H |-> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) )
7		riotaex 6615	. . 3 |- ( iota_ x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) e. _V
8	5, 6, 7	fvmpt 6282	. 2 |- ( A e. ~H -> ( ( z e. ~H |-> ( iota_ x e. H E. y e. ( _|_ ` H ) z = ( x +h y ) ) ) ` A ) = ( iota_ x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) )
9	2, 8	sylan9eq 2676	1 |- ( ( H e. CH /\ A e. ~H ) -> ( ( proj h ` H ) ` A ) = ( iota_ x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   ~Hchil 27776   +h cva 27777   CHcch 27786   _|_cort 27787   proj hcpjh 27794

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-pjh 28254

This theorem is referenced by:  pjpreeq  28257