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Theorem suppofss1d 6725
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 5556 . . . . . . . 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 5556 . . . . . . . 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 3556 . . . . . . 7  |-  ( A  i^i  A )  =  A
9 eqidd 2442 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2442 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( G `  y ) )
113, 6, 7, 7, 8, 9, 10ofval 6328 . . . . . 6  |-  ( (
ph  /\  y  e.  A )  ->  (
( F  oF X G ) `  y )  =  ( ( F `  y
) X ( G `
 y ) ) )
1211adantr 462 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  (
( F  oF X G ) `  y )  =  ( ( F `  y
) X ( G `
 y ) ) )
13 simpr 458 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  ( F `  y )  =  Z )
1413oveq1d 6105 . . . . 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 2797 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( Z X x )  =  Z )
1716adantr 462 . . . . . . 7  |-  ( (
ph  /\  y  e.  A )  ->  A. x  e.  B  ( Z X x )  =  Z )
184ffvelrnda 5840 . . . . . . . 8  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  e.  B )
19 simpr 458 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  ->  x  =  ( G `  y ) )
2019oveq2d 6106 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  -> 
( Z X x )  =  ( Z X ( G `  y ) ) )
2120eqeq1d 2449 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  A )  /\  x  =  ( G `  y ) )  -> 
( ( Z X x )  =  Z  <-> 
( Z X ( G `  y ) )  =  Z ) )
2218, 21rspcdv 3073 . . . . . . 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 462 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  ( F `  y )  =  Z )  ->  ( Z X ( G `  y ) )  =  Z )
2512, 14, 243eqtrd 2477 . . . 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 2797 . 2  |-  ( ph  ->  A. y  e.  A  ( ( F `  y )  =  Z  ->  ( ( F  oF X G ) `  y )  =  Z ) )
283, 6, 7, 7, 8offn 6330 . . 3  |-  ( ph  ->  ( F  oF X G )  Fn  A )
29 ssid 3372 . . . 4  |-  A  C_  A
3029a1i 11 . . 3  |-  ( ph  ->  A  C_  A )
31 suppofssd.2 . . 3  |-  ( ph  ->  Z  e.  B )
32 suppfnss 6713 . . 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 1220 . 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 1364    e. wcel 1761   A.wral 2713    C_ wss 3325    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   supp csupp 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-supp 6690
This theorem is referenced by:  frlmphllem  18164  rrxcph  20855
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