Theorem funresdfunsnss 5735

Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)

Assertion

Ref	Expression
funresdfunsnss	|- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Proof of Theorem funresdfunsnss

Step	Hyp	Ref	Expression
1		funrel 5248	. . . . 5 |- ( Fun F -> Rel F )
2		resdmdfsn 4965	. . . . 5 |- ( Rel F -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
3	1, 2	syl 14	. . . 4 |- ( Fun F -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
4	3	adantr 276	. . 3 |- ( ( Fun F /\ X e. dom F ) -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
5	4	uneq1d 3303	. 2 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
6		funfn 5261	. . 3 |- ( Fun F <-> F Fn dom F )
7		fnsnsplitss 5731	. . 3 |- ( ( F Fn dom F /\ X e. dom F ) -> ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
8	6, 7	sylanb 284	. 2 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
9	5, 8	eqsstrd 3206	1 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   \ cdif 3141   u. cun 3142   C_ wss 3144   {csn 3607   <.cop 3610   dom cdm 4641   |` cres 4643   Rel wrel 4646   Fun wfun 5225   Fn wfn 5226   ` cfv 5231

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239

This theorem is referenced by: (None)