Theorem funresdfunsnss 5887

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 5369	. . . . 5 |- ( Fun F -> Rel F )
2		resdmdfsn 5081	. . . . 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 3372	. 2 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = ( ( F |` ( dom F \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
6		funfn 5382	. . 3 |- ( Fun F <-> F Fn dom F )
7		fnsnsplitss 5883	. . 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 3274	1 |- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   \ cdif 3208   u. cun 3209   C_ wss 3211   {csn 3689   <.cop 3692   dom cdm 4749   |` cres 4751   Rel wrel 4754   Fun wfun 5346   Fn wfn 5347   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by: (None)