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Theorem suppofss1d 6949
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 )
suppofss1d.5  |-  ( (
ph  /\  x  e.  B )  ->  ( Z X x )  =  Z )
Assertion
Ref Expression
suppofss1d  |-  ( 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
Allowed substitution hint:    V( x)

Proof of Theorem suppofss1d
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 suppofssd.3 . . . . . . . 8  |-  ( ph  ->  F : A --> B )
2 ffn 5737 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
4 suppofssd.4 . . . . . . . 8  |-  ( ph  ->  G : A --> B )
5 ffn 5737 . . . . . . . 8  |-  ( G : A --> B  ->  G  Fn  A )
64, 5syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  A )
7 suppofssd.1 . . . . . . 7  |-  ( ph  ->  A  e.  V )
8 inidm 3712 . . . . . . 7  |-  ( A  i^i  A )  =  A
9 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( G `  y ) )
113, 6, 7, 7, 8, 9, 10ofval 6544 . . . . . 6  |-  ( (
ph  /\  y  e.  A )  ->  (
( F  oF X G ) `  y )  =  ( ( F `  y
) X ( G `
 y ) ) )
1211adantr 465 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  (
( F  oF X G ) `  y )  =  ( ( F `  y
) X ( G `
 y ) ) )
13 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  ( F `  y )  =  Z )
1413oveq1d 6310 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  (
( F `  y
) X ( G `
 y ) )  =  ( Z X ( G `  y
) ) )
15 suppofss1d.5 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( Z X x )  =  Z )
1615ralrimiva 2881 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( Z X x )  =  Z )
1716adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  A. x  e.  B  ( Z X x )  =  Z )
184ffvelrnda 6032 . . . . . . . 8  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  e.  B )
19 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  ->  x  =  ( G `  y ) )
2019oveq2d 6311 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  -> 
( Z X x )  =  ( Z X ( G `  y ) ) )
2120eqeq1d 2469 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  -> 
( ( Z X x )  =  Z  <-> 
( Z X ( G `  y ) )  =  Z ) )
2218, 21rspcdv 3222 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  ( A. x  e.  B  ( Z X x )  =  Z  ->  ( Z X ( G `  y ) )  =  Z ) )
2317, 22mpd 15 . . . . . 6  |-  ( (
ph  /\  y  e.  A )  ->  ( Z X ( G `  y ) )  =  Z )
2423adantr 465 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  ( Z X ( G `  y ) )  =  Z )
2512, 14, 243eqtrd 2512 . . . 4  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  (
( F  oF X G ) `  y )  =  Z )
2625ex 434 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  (
( F `  y
)  =  Z  -> 
( ( F  oF X G ) `
 y )  =  Z ) )
2726ralrimiva 2881 . 2  |-  ( ph  ->  A. y  e.  A  ( ( F `  y )  =  Z  ->  ( ( F  oF X G ) `  y )  =  Z ) )
283, 6, 7, 7, 8offn 6546 . . 3  |-  ( ph  ->  ( F  oF X G )  Fn  A )
29 ssid 3528 . . . 4  |-  A  C_  A
3029a1i 11 . . 3  |-  ( ph  ->  A  C_  A )
31 suppofssd.2 . . 3  |-  ( ph  ->  Z  e.  B )
32 suppfnss 6937 . . 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 ) ) )
3328, 3, 30, 7, 31, 32syl23anc 1235 . 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 ) ) )
3427, 33mpd 15 1  |-  ( ph  ->  ( ( F  oF X G ) supp 
Z )  C_  ( F supp  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   supp csupp 6913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-supp 6914
This theorem is referenced by:  frlmphllem  18680  rrxcph  21692
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