Theorem funresdfunsnss 5699

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 5215	. . . . 5 |- ( Fun F -> Rel F )
2		resdmdfsn 4934	. . . . 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 274	. . 3 |- ( ( Fun F /\ X e. dom F ) -> ( F |` ( _V \ { X } ) ) = ( F |` ( dom F \ { X } ) ) )
5	4	uneq1d 3280	. 2 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
6		funfn 5228	. . 3 |- ( Fun F <-> F Fn dom F )
7		fnsnsplitss 5695	. . 3 |- ( ( F Fn dom F /\ X e. dom F ) -> ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
8	6, 7	sylanb 282	. 2 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
9	5, 8	eqsstrd 3183	1 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   \ cdif 3118   u. cun 3119   C_ wss 3121   {csn 3583   <.cop 3586   dom cdm 4611   |` cres 4613   Rel wrel 4616   Fun wfun 5192   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by: (None)