Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppofss1d Structured version   Unicode version

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
suppofssd.2
suppofssd.3
suppofssd.4
suppofss1d.5
Assertion
Ref Expression
suppofss1d supp supp
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem suppofss1d
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 suppofssd.3 . . . . . . . 8
2 ffn 5737 . . . . . . . 8
31, 2syl 16 . . . . . . 7
4 suppofssd.4 . . . . . . . 8
5 ffn 5737 . . . . . . . 8
64, 5syl 16 . . . . . . 7
7 suppofssd.1 . . . . . . 7
8 inidm 3712 . . . . . . 7
9 eqidd 2468 . . . . . . 7
10 eqidd 2468 . . . . . . 7
113, 6, 7, 7, 8, 9, 10ofval 6544 . . . . . 6
1211adantr 465 . . . . 5
13 simpr 461 . . . . . 6
1413oveq1d 6310 . . . . 5
15 suppofss1d.5 . . . . . . . . 9
1615ralrimiva 2881 . . . . . . . 8
1716adantr 465 . . . . . . 7
184ffvelrnda 6032 . . . . . . . 8
19 simpr 461 . . . . . . . . . 10
2019oveq2d 6311 . . . . . . . . 9
2120eqeq1d 2469 . . . . . . . 8
2218, 21rspcdv 3222 . . . . . . 7
2317, 22mpd 15 . . . . . 6
2423adantr 465 . . . . 5
2512, 14, 243eqtrd 2512 . . . 4
2625ex 434 . . 3
2726ralrimiva 2881 . 2
283, 6, 7, 7, 8offn 6546 . . 3
29 ssid 3528 . . . 4
3029a1i 11 . . 3
31 suppofssd.2 . . 3
32 suppfnss 6937 . . 3 supp supp
3328, 3, 30, 7, 31, 32syl23anc 1235 . 2 supp supp
3427, 33mpd 15 1 supp supp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  wral 2817   wss 3481   wfn 5589  wf 5590  cfv 5594  (class class class)co 6295   cof 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
 Copyright terms: Public domain W3C validator