Theorem fdmdifeqresdif 42120

Description: The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)

Hypothesis

Ref	Expression
fdmdifeqresdif.f	|- F = ( x e. D |-> if ( x = Y , X , ( G ` x ) ) )

Assertion

Ref	Expression
fdmdifeqresdif	|- ( G : ( D \ { Y } ) --> R -> G = ( F |` ( D \ { Y } ) ) )

Distinct variable groups:   x, D   x, G   x, R   x, Y

Allowed substitution hints:   F( x)   X( x)

Proof of Theorem fdmdifeqresdif

Step	Hyp	Ref	Expression
1		eldifsn 4317	. . . . . 6 |- ( x e. ( D \ { Y } ) <-> ( x e. D /\ x =/= Y ) )
2		neneq 2800	. . . . . 6 |- ( x =/= Y -> -. x = Y )
3	1, 2	simplbiim 659	. . . . 5 |- ( x e. ( D \ { Y } ) -> -. x = Y )
4	3	adantl 482	. . . 4 |- ( ( G : ( D \ { Y } ) --> R /\ x e. ( D \ { Y } ) ) -> -. x = Y )
5	4	iffalsed 4097	. . 3 |- ( ( G : ( D \ { Y } ) --> R /\ x e. ( D \ { Y } ) ) -> if ( x = Y , X , ( G ` x ) ) = ( G ` x ) )
6	5	mpteq2dva 4744	. 2 |- ( G : ( D \ { Y } ) --> R -> ( x e. ( D \ { Y } ) |-> if ( x = Y , X , ( G ` x ) ) ) = ( x e. ( D \ { Y } ) |-> ( G ` x ) ) )
7		fdmdifeqresdif.f	. . . 4 |- F = ( x e. D |-> if ( x = Y , X , ( G ` x ) ) )
8	7	reseq1i 5392	. . 3 |- ( F |` ( D \ { Y } ) ) = ( ( x e. D |-> if ( x = Y , X , ( G ` x ) ) ) |` ( D \ { Y } ) )
9		difssd 3738	. . . 4 |- ( G : ( D \ { Y } ) --> R -> ( D \ { Y } ) C_ D )
10	9	resmptd 5452	. . 3 |- ( G : ( D \ { Y } ) --> R -> ( ( x e. D |-> if ( x = Y , X , ( G ` x ) ) ) |` ( D \ { Y } ) ) = ( x e. ( D \ { Y } ) |-> if ( x = Y , X , ( G ` x ) ) ) )
11	8, 10	syl5eq 2668	. 2 |- ( G : ( D \ { Y } ) --> R -> ( F |` ( D \ { Y } ) ) = ( x e. ( D \ { Y } ) |-> if ( x = Y , X , ( G ` x ) ) ) )
12		ffn 6045	. . 3 |- ( G : ( D \ { Y } ) --> R -> G Fn ( D \ { Y } ) )
13		dffn5 6241	. . 3 |- ( G Fn ( D \ { Y } ) <-> G = ( x e. ( D \ { Y } ) |-> ( G ` x ) ) )
14	12, 13	sylib 208	. 2 |- ( G : ( D \ { Y } ) --> R -> G = ( x e. ( D \ { Y } ) |-> ( G ` x ) ) )
15	6, 11, 14	3eqtr4rd 2667	1 |- ( G : ( D \ { Y } ) --> R -> G = ( F |` ( D \ { Y } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   ifcif 4086   {csn 4177   |-> cmpt 4729   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888

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-ne 2795  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  lincext2  42244  lincext3  42245