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Theorem suppss2f 29304
 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 2761 . . . 4 𝑙𝐴
3 nfcv 2761 . . . 4 𝑙𝐵
4 nfcsb1v 3534 . . . 4 𝑘𝑙 / 𝑘𝐵
5 csbeq1a 3527 . . . 4 (𝑘 = 𝑙𝐵 = 𝑙 / 𝑘𝐵)
61, 2, 3, 4, 5cbvmptf 4713 . . 3 (𝑘𝐴𝐵) = (𝑙𝐴𝑙 / 𝑘𝐵)
76oveq1i 6620 . 2 ((𝑘𝐴𝐵) supp 𝑍) = ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍)
8 suppss2f.n . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
98sbt 2418 . . . 4 [𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)
10 sbim 2394 . . . . 5 ([𝑙 / 𝑘]((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍) ↔ ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) → [𝑙 / 𝑘]𝐵 = 𝑍))
11 sban 2398 . . . . . . 7 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)))
12 suppss2f.p . . . . . . . . 9 𝑘𝜑
1312sbf 2379 . . . . . . . 8 ([𝑙 / 𝑘]𝜑𝜑)
14 suppss2f.w . . . . . . . . . 10 𝑘𝑊
151, 14nfdif 3714 . . . . . . . . 9 𝑘(𝐴𝑊)
1615clelsb3f 29191 . . . . . . . 8 ([𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊) ↔ 𝑙 ∈ (𝐴𝑊))
1713, 16anbi12i 732 . . . . . . 7 (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
1811, 17bitri 264 . . . . . 6 ([𝑙 / 𝑘](𝜑𝑘 ∈ (𝐴𝑊)) ↔ (𝜑𝑙 ∈ (𝐴𝑊)))
19 sbsbc 3425 . . . . . . 7 ([𝑙 / 𝑘]𝐵 = 𝑍[𝑙 / 𝑘]𝐵 = 𝑍)
20 vex 3192 . . . . . . . 8 𝑙 ∈ V
21 sbceq1g 3965 . . . . . . . 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 340 . . . . 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 7281 . 2 (𝜑 → ((𝑙𝐴𝑙 / 𝑘𝐵) supp 𝑍) ⊆ 𝑊)
297, 28syl5eqss 3633 1 (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  Ⅎwnf 1705  [wsb 1877   ∈ wcel 1987  Ⅎwnfc 2748  Vcvv 3189  [wsbc 3421  ⦋csb 3518   ∖ cdif 3556   ⊆ wss 3559   ↦ cmpt 4678  (class class class)co 6610   supp csupp 7247 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-supp 7248 This theorem is referenced by:  esumss  29939
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