Theorem fnsuppres 7322

Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)

Assertion

Ref	Expression
fnsuppres	|- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( ( F supp Z ) C_ A <-> ( F |` B ) = ( B X. { Z } ) ) )

Proof of Theorem fnsuppres

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fndm 5990	. . . . . 6 |- ( F Fn ( A u. B ) -> dom F = ( A u. B ) )
2		rabeq 3192	. . . . . 6 |- ( dom F = ( A u. B ) -> { a e. dom F | ( F ` a ) =/= Z } = { a e. ( A u. B ) | ( F ` a ) =/= Z } )
3	1, 2	syl 17	. . . . 5 |- ( F Fn ( A u. B ) -> { a e. dom F | ( F ` a ) =/= Z } = { a e. ( A u. B ) | ( F ` a ) =/= Z } )
4	3	3ad2ant1 1082	. . . 4 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> { a e. dom F | ( F ` a ) =/= Z } = { a e. ( A u. B ) | ( F ` a ) =/= Z } )
5	4	sseq1d 3632	. . 3 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( { a e. dom F | ( F ` a ) =/= Z } C_ A <-> { a e. ( A u. B ) | ( F ` a ) =/= Z } C_ A ) )
6		unss 3787	. . . . 5 |- ( ( { a e. A | ( F ` a ) =/= Z } C_ A /\ { a e. B | ( F ` a ) =/= Z } C_ A ) <-> ( { a e. A | ( F ` a ) =/= Z } u. { a e. B | ( F ` a ) =/= Z } ) C_ A )
7		ssrab2 3687	. . . . . 6 |- { a e. A | ( F ` a ) =/= Z } C_ A
8	7	biantrur 527	. . . . 5 |- ( { a e. B | ( F ` a ) =/= Z } C_ A <-> ( { a e. A | ( F ` a ) =/= Z } C_ A /\ { a e. B | ( F ` a ) =/= Z } C_ A ) )
9		rabun2 3906	. . . . . 6 |- { a e. ( A u. B ) | ( F ` a ) =/= Z } = ( { a e. A | ( F ` a ) =/= Z } u. { a e. B | ( F ` a ) =/= Z } )
10	9	sseq1i 3629	. . . . 5 |- ( { a e. ( A u. B ) | ( F ` a ) =/= Z } C_ A <-> ( { a e. A | ( F ` a ) =/= Z } u. { a e. B | ( F ` a ) =/= Z } ) C_ A )
11	6, 8, 10	3bitr4ri 293	. . . 4 |- ( { a e. ( A u. B ) | ( F ` a ) =/= Z } C_ A <-> { a e. B | ( F ` a ) =/= Z } C_ A )
12		rabss 3679	. . . . 5 |- ( { a e. B | ( F ` a ) =/= Z } C_ A <-> A. a e. B ( ( F ` a ) =/= Z -> a e. A ) )
13		fvres 6207	. . . . . . . . 9 |- ( a e. B -> ( ( F |` B ) ` a ) = ( F ` a ) )
14	13	adantl 482	. . . . . . . 8 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( ( F |` B ) ` a ) = ( F ` a ) )
15		simp2r 1088	. . . . . . . . 9 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> Z e. V )
16		fvconst2g 6467	. . . . . . . . 9 |- ( ( Z e. V /\ a e. B ) -> ( ( B X. { Z } ) ` a ) = Z )
17	15, 16	sylan 488	. . . . . . . 8 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( ( B X. { Z } ) ` a ) = Z )
18	14, 17	eqeq12d 2637	. . . . . . 7 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) <-> ( F ` a ) = Z ) )
19		nne 2798	. . . . . . . 8 |- ( -. ( F ` a ) =/= Z <-> ( F ` a ) = Z )
20	19	a1i 11	. . . . . . 7 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( -. ( F ` a ) =/= Z <-> ( F ` a ) = Z ) )
21		id 22	. . . . . . . . 9 |- ( a e. B -> a e. B )
22		simp3 1063	. . . . . . . . 9 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
23		minel 4033	. . . . . . . . 9 |- ( ( a e. B /\ ( A i^i B ) = (/) ) -> -. a e. A )
24	21, 22, 23	syl2anr 495	. . . . . . . 8 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> -. a e. A )
25		mtt 354	. . . . . . . 8 |- ( -. a e. A -> ( -. ( F ` a ) =/= Z <-> ( ( F ` a ) =/= Z -> a e. A ) ) )
26	24, 25	syl 17	. . . . . . 7 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( -. ( F ` a ) =/= Z <-> ( ( F ` a ) =/= Z -> a e. A ) ) )
27	18, 20, 26	3bitr2rd 297	. . . . . 6 |- ( ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) /\ a e. B ) -> ( ( ( F ` a ) =/= Z -> a e. A ) <-> ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
28	27	ralbidva 2985	. . . . 5 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( A. a e. B ( ( F ` a ) =/= Z -> a e. A ) <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
29	12, 28	syl5bb 272	. . . 4 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( { a e. B | ( F ` a ) =/= Z } C_ A <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
30	11, 29	syl5bb 272	. . 3 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( { a e. ( A u. B ) | ( F ` a ) =/= Z } C_ A <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
31	5, 30	bitrd 268	. 2 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( { a e. dom F | ( F ` a ) =/= Z } C_ A <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
32		fnfun 5988	. . . . . . 7 |- ( F Fn ( A u. B ) -> Fun F )
33	32	3anim1i 1248	. . . . . 6 |- ( ( F Fn ( A u. B ) /\ F e. W /\ Z e. V ) -> ( Fun F /\ F e. W /\ Z e. V ) )
34	33	3expb 1266	. . . . 5 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) ) -> ( Fun F /\ F e. W /\ Z e. V ) )
35		suppval1 7301	. . . . 5 |- ( ( Fun F /\ F e. W /\ Z e. V ) -> ( F supp Z ) = { a e. dom F | ( F ` a ) =/= Z } )
36	34, 35	syl 17	. . . 4 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) ) -> ( F supp Z ) = { a e. dom F | ( F ` a ) =/= Z } )
37	36	3adant3 1081	. . 3 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( F supp Z ) = { a e. dom F | ( F ` a ) =/= Z } )
38	37	sseq1d 3632	. 2 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( ( F supp Z ) C_ A <-> { a e. dom F | ( F ` a ) =/= Z } C_ A ) )
39		simp1 1061	. . . 4 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> F Fn ( A u. B ) )
40		ssun2 3777	. . . . 5 |- B C_ ( A u. B )
41	40	a1i 11	. . . 4 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> B C_ ( A u. B ) )
42		fnssres 6004	. . . 4 |- ( ( F Fn ( A u. B ) /\ B C_ ( A u. B ) ) -> ( F |` B ) Fn B )
43	39, 41, 42	syl2anc 693	. . 3 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( F |` B ) Fn B )
44		fnconstg 6093	. . . . 5 |- ( Z e. V -> ( B X. { Z } ) Fn B )
45	44	adantl 482	. . . 4 |- ( ( F e. W /\ Z e. V ) -> ( B X. { Z } ) Fn B )
46	45	3ad2ant2 1083	. . 3 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( B X. { Z } ) Fn B )
47		eqfnfv 6311	. . 3 |- ( ( ( F |` B ) Fn B /\ ( B X. { Z } ) Fn B ) -> ( ( F |` B ) = ( B X. { Z } ) <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
48	43, 46, 47	syl2anc 693	. 2 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( ( F |` B ) = ( B X. { Z } ) <-> A. a e. B ( ( F |` B ) ` a ) = ( ( B X. { Z } ) ` a ) ) )
49	31, 38, 48	3bitr4d 300	1 |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V ) /\ ( A i^i B ) = (/) ) -> ( ( F supp Z ) C_ A <-> ( F |` B ) = ( B X. { Z } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295

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  ax-un 6949

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-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  fnsuppeq0  7323  frlmsslss2  20114  resf1o  29505