MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsuppfn Structured version   Visualization version   GIF version

Theorem elsuppfn 7829
Description: An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
Assertion
Ref Expression
elsuppfn ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍)))

Proof of Theorem elsuppfn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 suppvalfn 7828 . . 3 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍})
21eleq2d 2903 . 2 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ 𝑆 ∈ {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍}))
3 fveq2 6667 . . . 4 (𝑖 = 𝑆 → (𝐹𝑖) = (𝐹𝑆))
43neeq1d 3080 . . 3 (𝑖 = 𝑆 → ((𝐹𝑖) ≠ 𝑍 ↔ (𝐹𝑆) ≠ 𝑍))
54elrab 3684 . 2 (𝑆 ∈ {𝑖𝑋 ∣ (𝐹𝑖) ≠ 𝑍} ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍))
62, 5syl6bb 288 1 ((𝐹 Fn 𝑋𝑋𝑉𝑍𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆𝑋 ∧ (𝐹𝑆) ≠ 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  {crab 3147   Fn wfn 6347  cfv 6352  (class class class)co 7148   supp csupp 7821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-supp 7822
This theorem is referenced by:  fvn0elsupp  7837  fvn0elsuppb  7838  rexsupp  7839  suppss  7851  suppssr  7852  suppofssd  7858  suppcofnd  7862  wemapso2lem  9005  cantnfle  9123  cantnfp1lem2  9131  cantnfp1lem3  9132  cantnfp1  9133  cantnflem1a  9137  cantnflem3  9143  cnfcomlem  9151  cnfcom3  9156  suppssfz  13352  ciclcl  17062  cicrcl  17063  mdegleb  24573  fsuppcurry1  30374  fsuppcurry2  30375  fdivmptf  44433  refdivmptf  44434
  Copyright terms: Public domain W3C validator