Theorem fvunsn 6445

Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Assertion

Ref	Expression
fvunsn	|- ( B =/= D -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) )

Proof of Theorem fvunsn

Step	Hyp	Ref	Expression
1		resundir 5411	. . . 4 |- ( ( A u. { <. B , C >. } ) |` { D } ) = ( ( A |` { D } ) u. ( { <. B , C >. } |` { D } ) )
2		nelsn 4212	. . . . . . 7 |- ( B =/= D -> -. B e. { D } )
3		ressnop0 6420	. . . . . . 7 |- ( -. B e. { D } -> ( { <. B , C >. } |` { D } ) = (/) )
4	2, 3	syl 17	. . . . . 6 |- ( B =/= D -> ( { <. B , C >. } |` { D } ) = (/) )
5	4	uneq2d 3767	. . . . 5 |- ( B =/= D -> ( ( A |` { D } ) u. ( { <. B , C >. } |` { D } ) ) = ( ( A |` { D } ) u. (/) ) )
6		un0 3967	. . . . 5 |- ( ( A |` { D } ) u. (/) ) = ( A |` { D } )
7	5, 6	syl6eq 2672	. . . 4 |- ( B =/= D -> ( ( A |` { D } ) u. ( { <. B , C >. } |` { D } ) ) = ( A |` { D } ) )
8	1, 7	syl5eq 2668	. . 3 |- ( B =/= D -> ( ( A u. { <. B , C >. } ) |` { D } ) = ( A |` { D } ) )
9	8	fveq1d 6193	. 2 |- ( B =/= D -> ( ( ( A u. { <. B , C >. } ) |` { D } ) ` D ) = ( ( A |` { D } ) ` D ) )
10		fvressn 6429	. . 3 |- ( D e. _V -> ( ( ( A u. { <. B , C >. } ) |` { D } ) ` D ) = ( ( A u. { <. B , C >. } ) ` D ) )
11		fvprc 6185	. . . 4 |- ( -. D e. _V -> ( ( ( A u. { <. B , C >. } ) |` { D } ) ` D ) = (/) )
12		fvprc 6185	. . . 4 |- ( -. D e. _V -> ( ( A u. { <. B , C >. } ) ` D ) = (/) )
13	11, 12	eqtr4d 2659	. . 3 |- ( -. D e. _V -> ( ( ( A u. { <. B , C >. } ) |` { D } ) ` D ) = ( ( A u. { <. B , C >. } ) ` D ) )
14	10, 13	pm2.61i 176	. 2 |- ( ( ( A u. { <. B , C >. } ) |` { D } ) ` D ) = ( ( A u. { <. B , C >. } ) ` D )
15		fvressn 6429	. . 3 |- ( D e. _V -> ( ( A |` { D } ) ` D ) = ( A ` D ) )
16		fvprc 6185	. . . 4 |- ( -. D e. _V -> ( ( A |` { D } ) ` D ) = (/) )
17		fvprc 6185	. . . 4 |- ( -. D e. _V -> ( A ` D ) = (/) )
18	16, 17	eqtr4d 2659	. . 3 |- ( -. D e. _V -> ( ( A |` { D } ) ` D ) = ( A ` D ) )
19	15, 18	pm2.61i 176	. 2 |- ( ( A |` { D } ) ` D ) = ( A ` D )
20	9, 14, 19	3eqtr3g 2679	1 |- ( B =/= D -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   |` cres 5116   ` cfv 5888

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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvpr1  6456  fvpr1g  6458  fvpr2g  6459  fvtp1  6460  fvtp1g  6463  ac6sfi  8204  cats1un  13475  ruclem6  14964  ruclem7  14965  wlkp1lem5  26574  wlkp1lem6  26575  fnchoice  39188  nnsum4primeseven  41688  nnsum4primesevenALTV  41689