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Theorem suppss2f 29773
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 2912 . . . 4 𝑙𝐴
3 nfcv 2912 . . . 4 𝑙𝐵
4 nfcsb1v 3696 . . . 4 𝑘𝑙 / 𝑘𝐵
5 csbeq1a 3689 . . . 4 (𝑘 = 𝑙𝐵 = 𝑙 / 𝑘𝐵)
61, 2, 3, 4, 5cbvmptf 4880 . . 3 (𝑘𝐴𝐵) = (𝑙𝐴𝑙 / 𝑘𝐵)
76oveq1i 6802 . 2 ((𝑘𝐴𝐵) supp 𝑍) = ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍)
8 suppss2f.n . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
98sbt 2565 . . . 4 [𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
10 sbim 2541 . . . . 5 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11 sban 2545 . . . . . . 7 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)))
12 suppss2f.p . . . . . . . . 9 𝑘𝜑
1312sbf 2526 . . . . . . . 8 ([𝑙 / 𝑘]𝜑𝜑)
14 suppss2f.w . . . . . . . . . 10 𝑘𝑊
151, 14nfdif 3880 . . . . . . . . 9 𝑘(𝐴𝑊)
1615clelsb3f 2916 . . . . . . . 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 3589 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍[𝑙 / 𝑘]𝐵 = 𝑍)
20 vex 3352 . . . . . . . 8 𝑙 ∈ V
21 sbceq1g 4130 . . . . . . . 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 7480 . 2 (𝜑 → ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍) ⊆ 𝑊)
297, 28syl5eqss 3796 1 (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wnf 1855  [wsb 2048  wcel 2144  wnfc 2899  Vcvv 3349  [wsbc 3585  csb 3680  cdif 3718  wss 3721  cmpt 4861  (class class class)co 6792   supp csupp 7445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  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 6795  df-oprab 6796  df-mpt2 6797  df-supp 7446
This theorem is referenced by:  esumss  30468
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