Theorem suppinisegdc 6513

Description: Relation between the support ( F supp Z ) and the initial segment ( `' F " { Z } ). (Contributed by Thierry Arnoux, 25-Jun-2024.) (Revised by Jim Kingdon, 18-Oct-2026.)

Hypotheses

Ref	Expression
suppinisegdc.fun	|- ( ph -> F : A --> B )
suppinisegdc.f	|- ( ph -> F e. V )
suppinisegdc.z	|- ( ph -> Z e. B )
suppinisegdc.dc	|- ( ph -> A. x e. B A. y e. B DECID x = y )

Assertion

Ref	Expression
suppinisegdc	|- ( ph -> ( A \ ( F supp Z ) ) = ( `' F " { Z } ) )

Distinct variable groups:   x, F, y   y, Z   x, B, y

Allowed substitution hints:   ph( x, y)   A( x, y)   V( x, y)   Z( x)

Proof of Theorem suppinisegdc

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3229	. . . 4 |- ( w e. ( A \ ( F supp Z ) ) <-> ( w e. A /\ -. w e. ( F supp Z ) ) )
2		eqeq2 2248	. . . . . . . . 9 |- ( y = Z -> ( ( F ` w ) = y <-> ( F ` w ) = Z ) )
3	2	dcbid 850	. . . . . . . 8 |- ( y = Z -> (DECID ( F ` w ) = y <-> DECID ( F ` w ) = Z ) )
4		eqeq1 2245	. . . . . . . . . . 11 |- ( x = ( F ` w ) -> ( x = y <-> ( F ` w ) = y ) )
5	4	dcbid 850	. . . . . . . . . 10 |- ( x = ( F ` w ) -> (DECID x = y <-> DECID ( F ` w ) = y ) )
6	5	ralbidv 2550	. . . . . . . . 9 |- ( x = ( F ` w ) -> ( A. y e. B DECID x = y <-> A. y e. B DECID ( F ` w ) = y ) )
7		suppinisegdc.dc	. . . . . . . . . 10 |- ( ph -> A. x e. B A. y e. B DECID x = y )
8	7	adantr 276	. . . . . . . . 9 |- ( ( ph /\ w e. A ) -> A. x e. B A. y e. B DECID x = y )
9		suppinisegdc.fun	. . . . . . . . . 10 |- ( ph -> F : A --> B )
10	9	ffvelcdmda 5846	. . . . . . . . 9 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. B )
11	6, 8, 10	rspcdva 2934	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> A. y e. B DECID ( F ` w ) = y )
12		suppinisegdc.z	. . . . . . . . 9 |- ( ph -> Z e. B )
13	12	adantr 276	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> Z e. B )
14	3, 11, 13	rspcdva 2934	. . . . . . 7 |- ( ( ph /\ w e. A ) -> DECID ( F ` w ) = Z )
15		nnedc 2425	. . . . . . 7 |- (DECID ( F ` w ) = Z -> ( -. ( F ` w ) =/= Z <-> ( F ` w ) = Z ) )
16	14, 15	syl 14	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( -. ( F ` w ) =/= Z <-> ( F ` w ) = Z ) )
17	9	ffnd 5537	. . . . . . . . 9 |- ( ph -> F Fn A )
18		suppinisegdc.f	. . . . . . . . 9 |- ( ph -> F e. V )
19		elsuppfng 6486	. . . . . . . . 9 |- ( ( F Fn A /\ F e. V /\ Z e. B ) -> ( w e. ( F supp Z ) <-> ( w e. A /\ ( F ` w ) =/= Z ) ) )
20	17, 18, 12, 19	syl3anc 1278	. . . . . . . 8 |- ( ph -> ( w e. ( F supp Z ) <-> ( w e. A /\ ( F ` w ) =/= Z ) ) )
21	20	baibd 935	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( w e. ( F supp Z ) <-> ( F ` w ) =/= Z ) )
22	21	notbid 677	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( -. w e. ( F supp Z ) <-> -. ( F ` w ) =/= Z ) )
23	18	adantr 276	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> F e. V )
24		vex 2824	. . . . . . . 8 |- w e. _V
25		fvexg 5717	. . . . . . . 8 |- ( ( F e. V /\ w e. _V ) -> ( F ` w ) e. _V )
26	23, 24, 25	sylancl 417	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. _V )
27		elsng 3724	. . . . . . 7 |- ( ( F ` w ) e. _V -> ( ( F ` w ) e. { Z } <-> ( F ` w ) = Z ) )
28	26, 27	syl 14	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) e. { Z } <-> ( F ` w ) = Z ) )
29	16, 22, 28	3bitr4d 220	. . . . 5 |- ( ( ph /\ w e. A ) -> ( -. w e. ( F supp Z ) <-> ( F ` w ) e. { Z } ) )
30	29	pm5.32da 456	. . . 4 |- ( ph -> ( ( w e. A /\ -. w e. ( F supp Z ) ) <-> ( w e. A /\ ( F ` w ) e. { Z } ) ) )
31	1, 30	bitrid 192	. . 3 |- ( ph -> ( w e. ( A \ ( F supp Z ) ) <-> ( w e. A /\ ( F ` w ) e. { Z } ) ) )
32		elpreima 5831	. . . 4 |- ( F Fn A -> ( w e. ( `' F " { Z } ) <-> ( w e. A /\ ( F ` w ) e. { Z } ) ) )
33	17, 32	syl 14	. . 3 |- ( ph -> ( w e. ( `' F " { Z } ) <-> ( w e. A /\ ( F ` w ) e. { Z } ) ) )
34	31, 33	bitr4d 191	. 2 |- ( ph -> ( w e. ( A \ ( F supp Z ) ) <-> w e. ( `' F " { Z } ) ) )
35	34	eqrdv 2236	1 |- ( ph -> ( A \ ( F supp Z ) ) = ( `' F " { Z } ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   _Vcvv 2821   \ cdif 3217   {csn 3709   `'ccnv 4775   "cima 4779   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   supp csupp 6479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480

This theorem is used by:  funisfsuppdc  7298