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Theorem suppss2f 29779
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 2913 . . . 4 𝑙𝐴
3 nfcv 2913 . . . 4 𝑙𝐵
4 nfcsb1v 3698 . . . 4 𝑘𝑙 / 𝑘𝐵
5 csbeq1a 3691 . . . 4 (𝑘 = 𝑙𝐵 = 𝑙 / 𝑘𝐵)
61, 2, 3, 4, 5cbvmptf 4882 . . 3 (𝑘𝐴𝐵) = (𝑙𝐴𝑙 / 𝑘𝐵)
76oveq1i 6803 . 2 ((𝑘𝐴𝐵) supp 𝑍) = ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍)
8 suppss2f.n . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
98sbt 2566 . . . 4 [𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
10 sbim 2542 . . . . 5 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11 sban 2546 . . . . . . 7 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)))
12 suppss2f.p . . . . . . . . 9 𝑘𝜑
1312sbf 2527 . . . . . . . 8 ([𝑙 / 𝑘]𝜑𝜑)
14 suppss2f.w . . . . . . . . . 10 𝑘𝑊
151, 14nfdif 3882 . . . . . . . . 9 𝑘(𝐴𝑊)
1615clelsb3f 2917 . . . . . . . 8 ([𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊) ↔ 𝑙 ∈ (𝐴𝑊))
1713, 16anbi12i 604 . . . . . . 7 (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
1811, 17bitri 264 . . . . . 6 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
19 sbsbc 3591 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍[𝑙 / 𝑘]𝐵 = 𝑍)
20 vex 3354 . . . . . . . 8 𝑙 ∈ V
21 sbceq1g 4132 . . . . . . . 8 (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍))
2220, 21ax-mp 5 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍)
2319, 22bitri 264 . . . . . 6 ([𝑙 / 𝑘]𝐵 = 𝑍𝑙 / 𝑘𝐵 = 𝑍)
2418, 23imbi12i 339 . . . . 5 (([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍) ↔ ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍))
2510, 24bitri 264 . . . 4 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍))
269, 25mpbi 220 . . 3 ((𝜑𝑙 ∈ (𝐴𝑊)) → 𝑙 / 𝑘𝐵 = 𝑍)
27 suppss2f.v . . 3 (𝜑𝐴𝑉)
2826, 27suppss2 7481 . 2 (𝜑 → ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍) ⊆ 𝑊)
297, 28syl5eqss 3798 1 (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  [wsb 2049  wcel 2145  wnfc 2900  Vcvv 3351  [wsbc 3587  csb 3682  cdif 3720  wss 3723  cmpt 4863  (class class class)co 6793   supp csupp 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-supp 7447
This theorem is referenced by:  esumss  30474
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