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Theorem suppss2f 32417
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 2899 . . . 4 𝑙𝐴
3 nfcv 2899 . . . 4 𝑙𝐵
4 nfcsb1v 3915 . . . 4 𝑘𝑙 / 𝑘𝐵
5 csbeq1a 3904 . . . 4 (𝑘 = 𝑙𝐵 = 𝑙 / 𝑘𝐵)
61, 2, 3, 4, 5cbvmptf 5251 . . 3 (𝑘𝐴𝐵) = (𝑙𝐴𝑙 / 𝑘𝐵)
76oveq1i 7424 . 2 ((𝑘𝐴𝐵) supp 𝑍) = ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍)
8 suppss2f.n . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
98sbt 2062 . . . 4 [𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
10 sbim 2293 . . . . 5 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11 sban 2076 . . . . . . 7 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)))
12 suppss2f.p . . . . . . . . 9 𝑘𝜑
1312sbf 2258 . . . . . . . 8 ([𝑙 / 𝑘]𝜑𝜑)
14 suppss2f.w . . . . . . . . . 10 𝑘𝑊
151, 14nfdif 4121 . . . . . . . . 9 𝑘(𝐴𝑊)
1615clelsb1fw 2903 . . . . . . . 8 ([𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊) ↔ 𝑙 ∈ (𝐴𝑊))
1713, 16anbi12i 627 . . . . . . 7 (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
1811, 17bitri 275 . . . . . 6 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
19 sbsbc 3779 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍[𝑙 / 𝑘]𝐵 = 𝑍)
20 sbceq1g 4410 . . . . . . . 8 (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍))
2120elv 3476 . . . . . . 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 229 . . 3 ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍)
26 suppss2f.v . . 3 (𝜑𝐴𝑉)
2725, 26suppss2 8199 . 2 (𝜑 → ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍) ⊆ 𝑊)
287, 27eqsstrid 4026 1 (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wnf 1778  [wsb 2060  wcel 2099  wnfc 2879  Vcvv 3470  [wsbc 3775  csb 3890  cdif 3942  wss 3945  cmpt 5225  (class class class)co 7414   supp csupp 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-supp 8160
This theorem is referenced by:  elrspunidl  33138  esumss  33685
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