Theorem suppofss2d 7333

Description: Condition for the support of a function operation to be a subset of the support of the right 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 )
suppofss2d.5	|- ( ( ph /\ x e. B ) -> ( x X Z ) = Z )

Assertion

Ref	Expression
suppofss2d	|- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) )

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

Allowed substitution hint:   V( x)

Proof of Theorem suppofss2d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppofssd.3	. . . . . . . 8 |- ( ph -> F : A --> B )
2		ffn 6045	. . . . . . . 8 |- ( F : A --> B -> F Fn A )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> F Fn A )
4		suppofssd.4	. . . . . . . 8 |- ( ph -> G : A --> B )
5		ffn 6045	. . . . . . . 8 |- ( G : A --> B -> G Fn A )
6	4, 5	syl 17	. . . . . . 7 |- ( ph -> G Fn A )
7		suppofssd.1	. . . . . . 7 |- ( ph -> A e. V )
8		inidm 3822	. . . . . . 7 |- ( A i^i A ) = A
9		eqidd 2623	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
10		eqidd 2623	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( G ` y ) )
11	3, 6, 7, 7, 8, 9, 10	ofval 6906	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( ( F oF X G ) ` y ) = ( ( F ` y ) X ( G ` y ) ) )
12	11	adantr 481	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( G ` y ) = Z ) -> ( ( F oF X G ) ` y ) = ( ( F ` y ) X ( G ` y ) ) )
13		simpr 477	. . . . . 6 |- ( ( ( ph /\ y e. A ) /\ ( G ` y ) = Z ) -> ( G ` y ) = Z )
14	13	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( G ` y ) = Z ) -> ( ( F ` y ) X ( G ` y ) ) = ( ( F ` y ) X Z ) )
15		suppofss2d.5	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> ( x X Z ) = Z )
16	15	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. x e. B ( x X Z ) = Z )
17	16	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. A ) -> A. x e. B ( x X Z ) = Z )
18	1	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. B )
19		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ y e. A ) /\ x = ( F ` y ) ) -> x = ( F ` y ) )
20	19	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ph /\ y e. A ) /\ x = ( F ` y ) ) -> ( x X Z ) = ( ( F ` y ) X Z ) )
21	20	eqeq1d 2624	. . . . . . . 8 |- ( ( ( ph /\ y e. A ) /\ x = ( F ` y ) ) -> ( ( x X Z ) = Z <-> ( ( F ` y ) X Z ) = Z ) )
22	18, 21	rspcdv 3312	. . . . . . 7 |- ( ( ph /\ y e. A ) -> ( A. x e. B ( x X Z ) = Z -> ( ( F ` y ) X Z ) = Z ) )
23	17, 22	mpd 15	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( ( F ` y ) X Z ) = Z )
24	23	adantr 481	. . . . 5 |- ( ( ( ph /\ y e. A ) /\ ( G ` y ) = Z ) -> ( ( F ` y ) X Z ) = Z )
25	12, 14, 24	3eqtrd 2660	. . . 4 |- ( ( ( ph /\ y e. A ) /\ ( G ` y ) = Z ) -> ( ( F oF X G ) ` y ) = Z )
26	25	ex 450	. . 3 |- ( ( ph /\ y e. A ) -> ( ( G ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) )
27	26	ralrimiva 2966	. 2 |- ( ph -> A. y e. A ( ( G ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) )
28	3, 6, 7, 7, 8	offn 6908	. . 3 |- ( ph -> ( F oF X G ) Fn A )
29		ssid 3624	. . . 4 |- A C_ A
30	29	a1i 11	. . 3 |- ( ph -> A C_ A )
31		suppofssd.2	. . 3 |- ( ph -> Z e. B )
32		suppfnss 7320	. . 3 |- ( ( ( ( F oF X G ) Fn A /\ G Fn A ) /\ ( A C_ A /\ A e. V /\ Z e. B ) ) -> ( A. y e. A ( ( G ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) ) )
33	28, 6, 30, 7, 31, 32	syl23anc 1333	. 2 |- ( ph -> ( A. y e. A ( ( G ` y ) = Z -> ( ( F oF X G ) ` y ) = Z ) -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) ) )
34	27, 33	mpd 15	1 |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   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-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-of 6897  df-supp 7296

This theorem is referenced by:  frlmphl  20120