Theorem fundif 5317

Description: A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)

Assertion

Ref	Expression
fundif	|- ( Fun F -> Fun ( F \ A ) )

Proof of Theorem fundif

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldif 4794	. . 3 |- ( Rel F -> Rel ( F \ A ) )
2		brdif 4096	. . . . . . 7 |- ( x ( F \ A ) y <-> ( x F y /\ -. x A y ) )
3		brdif 4096	. . . . . . 7 |- ( x ( F \ A ) z <-> ( x F z /\ -. x A z ) )
4		pm2.27 40	. . . . . . . 8 |- ( ( x F y /\ x F z ) -> ( ( ( x F y /\ x F z ) -> y = z ) -> y = z ) )
5	4	ad2ant2r 509	. . . . . . 7 |- ( ( ( x F y /\ -. x A y ) /\ ( x F z /\ -. x A z ) ) -> ( ( ( x F y /\ x F z ) -> y = z ) -> y = z ) )
6	2, 3, 5	syl2anb 291	. . . . . 6 |- ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> ( ( ( x F y /\ x F z ) -> y = z ) -> y = z ) )
7	6	com12 30	. . . . 5 |- ( ( ( x F y /\ x F z ) -> y = z ) -> ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) )
8	7	alimi 1477	. . . 4 |- ( A. z ( ( x F y /\ x F z ) -> y = z ) -> A. z ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) )
9	8	2alimi 1478	. . 3 |- ( A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) -> A. x A. y A. z ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) )
10	1, 9	anim12i 338	. 2 |- ( ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) -> ( Rel ( F \ A ) /\ A. x A. y A. z ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) ) )
11		dffun2 5280	. 2 |- ( Fun F <-> ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) )
12		dffun2 5280	. 2 |- ( Fun ( F \ A ) <-> ( Rel ( F \ A ) /\ A. x A. y A. z ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) ) )
13	10, 11, 12	3imtr4i 201	1 |- ( Fun F -> Fun ( F \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1370   \ cdif 3162   class class class wbr 4043   Rel wrel 4679   Fun wfun 5264

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-rel 4681  df-cnv 4682  df-co 4683  df-fun 5272

This theorem is referenced by:  fundm2domnop  10989  fun2dmnop  10991