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Theorem suppss2f 32648
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 𝑘𝜑
suppss2f.a 𝑘𝐴
suppss2f.w 𝑘𝑊
suppss2f.n ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
suppss2f.v (𝜑𝐴𝑉)
Assertion
Ref Expression
suppss2f (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
Distinct variable group:   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem suppss2f
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 suppss2f.a . . . 4 𝑘𝐴
2 nfcv 2905 . . . 4 𝑙𝐴
3 nfcv 2905 . . . 4 𝑙𝐵
4 nfcsb1v 3923 . . . 4 𝑘𝑙 / 𝑘𝐵
5 csbeq1a 3913 . . . 4 (𝑘 = 𝑙𝐵 = 𝑙 / 𝑘𝐵)
61, 2, 3, 4, 5cbvmptf 5251 . . 3 (𝑘𝐴𝐵) = (𝑙𝐴𝑙 / 𝑘𝐵)
76oveq1i 7441 . 2 ((𝑘𝐴𝐵) supp 𝑍) = ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍)
8 suppss2f.n . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
98sbt 2066 . . . 4 [𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
10 sbim 2303 . . . . 5 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11 sban 2080 . . . . . . 7 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)))
12 suppss2f.p . . . . . . . . 9 𝑘𝜑
1312sbf 2271 . . . . . . . 8 ([𝑙 / 𝑘]𝜑𝜑)
14 suppss2f.w . . . . . . . . . 10 𝑘𝑊
151, 14nfdif 4129 . . . . . . . . 9 𝑘(𝐴𝑊)
1615clelsb1fw 2909 . . . . . . . 8 ([𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊) ↔ 𝑙 ∈ (𝐴𝑊))
1713, 16anbi12i 628 . . . . . . 7 (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
1811, 17bitri 275 . . . . . 6 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
19 sbsbc 3792 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍[𝑙 / 𝑘]𝐵 = 𝑍)
20 sbceq1g 4417 . . . . . . . 8 (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍))
2120elv 3485 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍)
2219, 21bitri 275 . . . . . 6 ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍)
2318, 22imbi12i 350 . . . . 5 (([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍))
2410, 23bitri 275 . . . 4 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍))
259, 24mpbi 230 . . 3 ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍)
26 suppss2f.v . . 3 (𝜑𝐴𝑉)
2725, 26suppss2 8225 . 2 (𝜑 → ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍) ⊆ 𝑊)
287, 27eqsstrid 4022 1 (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  [wsb 2064  wcel 2108  wnfc 2890  Vcvv 3480  [wsbc 3788  csb 3899  cdif 3948  wss 3951  cmpt 5225  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  elrspunidl  33456  esumss  34073
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