Theorem suppofss1dcl 6442

Description: Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)

Hypotheses

Ref	Expression
suppofssd.1	|- ( ph -> A e. V )
suppofssd.2	|- ( ph -> Z e. B )
suppofssd.3	|- ( ph -> F : A --> B )
suppofssd.4	|- ( ph -> G : A --> B )
suppofss1dcl.cl	|- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )
suppofss1d.5	|- ( ( ph /\ x e. B ) -> ( Z X x ) = Z )

Assertion

Ref	Expression
suppofss1dcl	|- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) )

Distinct variable groups:   x, A   x, B   x, F   x, G   x, X   x, Z   ph, x   u, B, v   ph, u, v   u, F, v   v, G   u, X, v

Allowed substitution hints:   A( v, u)   G( u)   V( x, v, u)   Z( v, u)

Proof of Theorem suppofss1dcl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 |- ( ph -> F : A --> B )
2	1	ffnd 5490	. . . . . . 7 |- ( ph -> F Fn A )
3		suppofssd.4	. . . . . . . 8 |- ( ph -> G : A --> B )
4	3	ffnd 5490	. . . . . . 7 |- ( ph -> G Fn A )
5		suppofssd.1	. . . . . . 7 |- ( ph -> A e. V )
6		inidm 3418	. . . . . . 7 |- ( A i^i A ) = A
7		eqidd 2232	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
8		eqidd 2232	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( G ` y ) )
9		oveq2 6036	. . . . . . . . 9 |- ( v = ( G ` y ) -> ( ( F ` y ) X v ) = ( ( F ` y ) X ( G ` y ) ) )
10	9	eleq1d 2300	. . . . . . . 8 |- ( v = ( G ` y ) -> ( ( ( F ` y ) X v ) e. B <-> ( ( F ` y ) X ( G ` y ) ) e. B ) )
11		oveq1 6035	. . . . . . . . . . 11 |- ( u = ( F ` y ) -> ( u X v ) = ( ( F ` y ) X v ) )
12	11	eleq1d 2300	. . . . . . . . . 10 |- ( u = ( F ` y ) -> ( ( u X v ) e. B <-> ( ( F ` y ) X v ) e. B ) )
13	12	ralbidv 2533	. . . . . . . . 9 |- ( u = ( F ` y ) -> ( A. v e. B ( u X v ) e. B <-> A. v e. B ( ( F ` y ) X v ) e. B ) )
14		suppofss1dcl.cl	. . . . . . . . . . 11 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )
15	14	ralrimivva 2615	. . . . . . . . . 10 |- ( ph -> A. u e. B A. v e. B ( u X v ) e. B )
16	15	adantr 276	. . . . . . . . 9 |- ( ( ph /\ y e. A ) -> A. u e. B A. v e. B ( u X v ) e. B )
17	1	ffvelcdmda 5790	. . . . . . . . 9 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. B )
18	13, 16, 17	rspcdva 2916	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> A. v e. B ( ( F ` y ) X v ) e. B )
19	3	ffvelcdmda 5790	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> ( G ` y ) e. B )
20	10, 18, 19	rspcdva 2916	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( ( F ` y ) X ( G ` y ) ) e. B )
21	2, 4, 5, 5, 6, 7, 8, 20	ofvalg 6254	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( ( F oF X G ) ` y ) = ( ( F ` y ) X ( G ` y ) ) )
22	21	adantr 276	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( F ` y ) = Z ) -> ( ( F oF X G ) ` y ) = ( ( F ` y ) X ( G ` y ) ) )
23		simpr 110	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ ( F ` y ) = Z ) -> ( F ` y ) = Z )
24	23	oveq1d 6043	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( F ` y ) = Z ) -> ( ( F ` y ) X ( G ` y ) ) = ( Z X ( G ` y ) ) )
25		suppofss1d.5	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> ( Z X x ) = Z )
26	25	ralrimiva 2606	. . . . . . . 8 |- ( ph -> A. x e. B ( Z X x ) = Z )
27	26	adantr 276	. . . . . . 7 |- ( ( ph /\ y e. A ) -> A. x e. B ( Z X x ) = Z )
28		simpr 110	. . . . . . . . . 10 |- ( ( ( ph /\ y e. A ) /\ x = ( G ` y ) ) -> x = ( G ` y ) )
29	28	oveq2d 6044	. . . . . . . . 9 |- ( ( ( ph /\ y e. A ) /\ x = ( G ` y ) ) -> ( Z X x ) = ( Z X ( G ` y ) ) )
30	29	eqeq1d 2240	. . . . . . . 8 |- ( ( ( ph /\ y e. A ) /\ x = ( G ` y ) ) -> ( ( Z X x ) = Z <-> ( Z X ( G ` y ) ) = Z ) )
31	19, 30	rspcdv 2914	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( A. x e. B ( Z X x ) = Z -> ( Z X ( G ` y ) ) = Z ) )
32	27, 31	mpd 13	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( Z X ( G ` y ) ) = Z )
33	32	adantr 276	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( F ` y ) = Z ) -> ( Z X ( G ` y ) ) = Z )
34	22, 24, 33	3eqtrd 2268	. . . 4 |- ( ( ( ph /\ y e. A ) /\ ( F ` y ) = Z ) -> ( ( F oF X G ) ` y ) = Z )
35	34	ex 115	. . 3 |- ( ( ph /\ y e. A ) -> ( ( F ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) )
36	35	ralrimiva 2606	. 2 |- ( ph -> A. y e. A ( ( F ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) )
37	14, 1, 3, 5, 5, 6	off 6257	. . . 4 |- ( ph -> ( F oF X G ) : A --> B )
38	37	ffnd 5490	. . 3 |- ( ph -> ( F oF X G ) Fn A )
39		ssidd 3249	. . 3 |- ( ph -> A C_ A )
40		suppofssd.2	. . 3 |- ( ph -> Z e. B )
41		suppfnss 6435	. . 3 |- ( ( ( ( F oF X G ) Fn A /\ F Fn A ) /\ ( A C_ A /\ A e. V /\ Z e. B ) ) -> ( A. y e. A ( ( F ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) ) )
42	38, 2, 39, 5, 40, 41	syl23anc 1281	. 2 |- ( ph -> ( A. y e. A ( ( F ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) ) )
43	36, 42	mpd 13	1 |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   oFcof 6242   supp csupp 6413

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-supp 6414

This theorem is referenced by: (None)