Theorem suppss3 29502

Description: Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
suppss3.1	|- G = ( x e. A |-> B )
suppss3.a	|- ( ph -> A e. V )
suppss3.z	|- ( ph -> Z e. W )
suppss3.2	|- ( ph -> F Fn A )
suppss3.3	|- ( ( ph /\ x e. A /\ ( F ` x ) = Z ) -> B = Z )

Assertion

Ref	Expression
suppss3	|- ( ph -> ( G supp Z ) C_ ( F supp Z ) )

Distinct variable groups:   x, A   x, F   x, Z   ph, x

Allowed substitution hints:   B( x)   G( x)   V( x)   W( x)

Proof of Theorem suppss3

Step	Hyp	Ref	Expression
1		suppss3.1	. . 3 |- G = ( x e. A |-> B )
2	1	oveq1i 6660	. 2 |- ( G supp Z ) = ( ( x e. A |-> B ) supp Z )
3		simpl 473	. . . 4 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ph )
4		eldifi 3732	. . . . 5 |- ( x e. ( A \ ( F supp Z ) ) -> x e. A )
5	4	adantl 482	. . . 4 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> x e. A )
6		suppss3.2	. . . . . . . . . . . . . 14 |- ( ph -> F Fn A )
7		suppss3.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. V )
8		fnex 6481	. . . . . . . . . . . . . 14 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
9	6, 7, 8	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> F e. _V )
10		suppss3.z	. . . . . . . . . . . . 13 |- ( ph -> Z e. W )
11		suppimacnv 7306	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
12	9, 10, 11	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
13	12	eleq2d 2687	. . . . . . . . . . 11 |- ( ph -> ( x e. ( F supp Z ) <-> x e. ( `' F " ( _V \ { Z } ) ) ) )
14		elpreima 6337	. . . . . . . . . . . 12 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
15	6, 14	syl 17	. . . . . . . . . . 11 |- ( ph -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
16	13, 15	bitrd 268	. . . . . . . . . 10 |- ( ph -> ( x e. ( F supp Z ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
17	16	baibd 948	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> ( x e. ( F supp Z ) <-> ( F ` x ) e. ( _V \ { Z } ) ) )
18	17	notbid 308	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( -. x e. ( F supp Z ) <-> -. ( F ` x ) e. ( _V \ { Z } ) ) )
19	18	biimpd 219	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( -. x e. ( F supp Z ) -> -. ( F ` x ) e. ( _V \ { Z } ) ) )
20	19	expimpd 629	. . . . . 6 |- ( ph -> ( ( x e. A /\ -. x e. ( F supp Z ) ) -> -. ( F ` x ) e. ( _V \ { Z } ) ) )
21		eldif 3584	. . . . . 6 |- ( x e. ( A \ ( F supp Z ) ) <-> ( x e. A /\ -. x e. ( F supp Z ) ) )
22		fvex 6201	. . . . . . . 8 |- ( F ` x ) e. _V
23		eldifsn 4317	. . . . . . . 8 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) )
24	22, 23	mpbiran 953	. . . . . . 7 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( F ` x ) =/= Z )
25	24	necon2bbii 2845	. . . . . 6 |- ( ( F ` x ) = Z <-> -. ( F ` x ) e. ( _V \ { Z } ) )
26	20, 21, 25	3imtr4g 285	. . . . 5 |- ( ph -> ( x e. ( A \ ( F supp Z ) ) -> ( F ` x ) = Z ) )
27	26	imp 445	. . . 4 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> ( F ` x ) = Z )
28		suppss3.3	. . . 4 |- ( ( ph /\ x e. A /\ ( F ` x ) = Z ) -> B = Z )
29	3, 5, 27, 28	syl3anc 1326	. . 3 |- ( ( ph /\ x e. ( A \ ( F supp Z ) ) ) -> B = Z )
30	29, 7	suppss2 7329	. 2 |- ( ph -> ( ( x e. A |-> B ) supp Z ) C_ ( F supp Z ) )
31	2, 30	syl5eqss 3649	1 |- ( ph -> ( G supp Z ) C_ ( F supp Z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   |-> cmpt 4729   `'ccnv 5113   "cima 5117   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-rep 4771  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-reu 2919  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  eulerpartlems  30422