Theorem fundif 5374

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 4847	. . 3 |- ( Rel F -> Rel ( F \ A ) )
2		brdif 4142	. . . . . . 7 |- ( x ( F \ A ) y <-> ( x F y /\ -. x A y ) )
3		brdif 4142	. . . . . . 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 1503	. . . 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 1504	. . 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 5336	. 2 |- ( Fun F <-> ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) )
12		dffun2 5336	. 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 1395   \ cdif 3197   class class class wbr 4088   Rel wrel 4730   Fun wfun 5320

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  fundm2domnop  11109  fun2dmnop  11111  edgstruct  15914