Theorem fundif 5935

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 5238	. . 3 |- ( Rel F -> Rel ( F \ A ) )
2		brdif 4705	. . . . . . 7 |- ( x ( F \ A ) y <-> ( x F y /\ -. x A y ) )
3		brdif 4705	. . . . . . 7 |- ( x ( F \ A ) z <-> ( x F z /\ -. x A z ) )
4		pm2.27 42	. . . . . . . 8 |- ( ( x F y /\ x F z ) -> ( ( ( x F y /\ x F z ) -> y = z ) -> y = z ) )
5	4	ad2ant2r 783	. . . . . . 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 496	. . . . . 6 |- ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> ( ( ( x F y /\ x F z ) -> y = z ) -> y = z ) )
7	6	com12 32	. . . . 5 |- ( ( ( x F y /\ x F z ) -> y = z ) -> ( ( x ( F \ A ) y /\ x ( F \ A ) z ) -> y = z ) )
8	7	alimi 1739	. . . 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 1740	. . 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 590	. 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 5898	. 2 |- ( Fun F <-> ( Rel F /\ A. x A. y A. z ( ( x F y /\ x F z ) -> y = z ) ) )
12		dffun2 5898	. 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 281	1 |- ( Fun F -> Fun ( F \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   \ cdif 3571   class class class wbr 4653   Rel wrel 5119   Fun wfun 5882

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-ral 2917  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  fundmge2nop  13275  fun2dmnop  13277  basvtxvalOLD  25903  edgfiedgvalOLD  25904