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Theorem suppss2f 28314
 Description: Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
Hypotheses
Ref Expression
suppss2f.p
suppss2f.a
suppss2f.w
suppss2f.n
suppss2f.v
Assertion
Ref Expression
suppss2f supp
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()   ()

Proof of Theorem suppss2f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 suppss2f.a . . . 4
2 nfcv 2612 . . . 4
3 nfcv 2612 . . . 4
4 nfcsb1v 3365 . . . 4
5 csbeq1a 3358 . . . 4
61, 2, 3, 4, 5cbvmptf 4486 . . 3
76oveq1i 6318 . 2 supp supp
8 suppss2f.n . . . . 5
98sbt 2268 . . . 4
10 sbim 2244 . . . . 5
11 sban 2248 . . . . . . 7
12 suppss2f.p . . . . . . . . 9
1312sbf 2229 . . . . . . . 8
14 suppss2f.w . . . . . . . . . 10
151, 14nfdif 3543 . . . . . . . . 9
1615clelsb3f 28195 . . . . . . . 8
1713, 16anbi12i 711 . . . . . . 7
1811, 17bitri 257 . . . . . 6
19 sbsbc 3259 . . . . . . 7
20 vex 3034 . . . . . . . 8
21 sbceq1g 3781 . . . . . . . 8
2220, 21ax-mp 5 . . . . . . 7
2319, 22bitri 257 . . . . . 6
2418, 23imbi12i 333 . . . . 5
2510, 24bitri 257 . . . 4
269, 25mpbi 213 . . 3
27 suppss2f.v . . 3
2826, 27suppss2 6968 . 2 supp
297, 28syl5eqss 3462 1 supp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452  wnf 1675  wsb 1805   wcel 1904  wnfc 2599  cvv 3031  wsbc 3255  csb 3349   cdif 3387   wss 3390   cmpt 4454  (class class class)co 6308   supp csupp 6933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-supp 6934 This theorem is referenced by:  esumss  28967
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