Theorem suppofss1dcl 6463

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 5508	. . . . . . 7 |- ( ph -> F Fn A )
3		suppofssd.4	. . . . . . . 8 |- ( ph -> G : A --> B )
4	3	ffnd 5508	. . . . . . 7 |- ( ph -> G Fn A )
5		suppofssd.1	. . . . . . 7 |- ( ph -> A e. V )
6		inidm 3429	. . . . . . 7 |- ( A i^i A ) = A
7		eqidd 2233	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
8		eqidd 2233	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( G ` y ) )
9		oveq2 6057	. . . . . . . . 9 |- ( v = ( G ` y ) -> ( ( F ` y ) X v ) = ( ( F ` y ) X ( G ` y ) ) )
10	9	eleq1d 2301	. . . . . . . 8 |- ( v = ( G ` y ) -> ( ( ( F ` y ) X v ) e. B <-> ( ( F ` y ) X ( G ` y ) ) e. B ) )
11		oveq1 6056	. . . . . . . . . . 11 |- ( u = ( F ` y ) -> ( u X v ) = ( ( F ` y ) X v ) )
12	11	eleq1d 2301	. . . . . . . . . 10 |- ( u = ( F ` y ) -> ( ( u X v ) e. B <-> ( ( F ` y ) X v ) e. B ) )
13	12	ralbidv 2542	. . . . . . . . 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 2624	. . . . . . . . . 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 5811	. . . . . . . . 9 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. B )
18	13, 16, 17	rspcdva 2925	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> A. v e. B ( ( F ` y ) X v ) e. B )
19	3	ffvelcdmda 5811	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> ( G ` y ) e. B )
20	10, 18, 19	rspcdva 2925	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( ( F ` y ) X ( G ` y ) ) e. B )
21	2, 4, 5, 5, 6, 7, 8, 20	ofvalg 6275	. . . . . 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 6064	. . . . 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 2615	. . . . . . . 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 6065	. . . . . . . . 9 |- ( ( ( ph /\ y e. A ) /\ x = ( G ` y ) ) -> ( Z X x ) = ( Z X ( G ` y ) ) )
30	29	eqeq1d 2241	. . . . . . . 8 |- ( ( ( ph /\ y e. A ) /\ x = ( G ` y ) ) -> ( ( Z X x ) = Z <-> ( Z X ( G ` y ) ) = Z ) )
31	19, 30	rspcdv 2923	. . . . . . 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 2269	. . . 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 2615	. 2 |- ( ph -> A. y e. A ( ( F ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) )
37	14, 1, 3, 5, 5, 6	off 6278	. . . 4 |- ( ph -> ( F oF X G ) : A --> B )
38	37	ffnd 5508	. . 3 |- ( ph -> ( F oF X G ) Fn A )
39		ssidd 3258	. . 3 |- ( ph -> A C_ A )
40		suppofssd.2	. . 3 |- ( ph -> Z e. B )
41		suppfnss 6456	. . 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 2203   A.wral 2520   C_ wss 3210   Fn wfn 5346   -->wf 5347   ` cfv 5351  (class class class)co 6049   oFcof 6263   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-supp 6435

This theorem is referenced by: (None)