Theorem fundif 5371

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 4845	. . 3 |- ( Rel F -> Rel ( F \ A ) )
2		brdif 4140	. . . . . . 7 |- ( x ( F \ A ) y <-> ( x F y /\ -. x A y ) )
3		brdif 4140	. . . . . . 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 1501	. . . 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 1502	. . 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 5334	. 2 |- ( Fun F <-> ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) )
12		dffun2 5334	. 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 1393   \ cdif 3195   class class class wbr 4086   Rel wrel 4728   Fun wfun 5318

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-rel 4730  df-cnv 4731  df-co 4732  df-fun 5326

This theorem is referenced by:  fundm2domnop  11100  fun2dmnop  11102  edgstruct  15905