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Theorem mptsuppdifd 8210
Description: The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
Hypotheses
Ref Expression
mptsuppdifd.f 𝐹 = (𝑥𝐴𝐵)
mptsuppdifd.a (𝜑𝐴𝑉)
mptsuppdifd.z (𝜑𝑍𝑊)
Assertion
Ref Expression
mptsuppdifd (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptsuppdifd
StepHypRef Expression
1 mptsuppdifd.f . . . 4 𝐹 = (𝑥𝐴𝐵)
2 mptsuppdifd.a . . . . 5 (𝜑𝐴𝑉)
32mptexd 7244 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ V)
41, 3eqeltrid 2843 . . 3 (𝜑𝐹 ∈ V)
5 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
6 suppimacnv 8198 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 584 . 2 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
81mptpreima 6260 . 2 (𝐹 “ (V ∖ {𝑍})) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})}
97, 8eqtrdi 2791 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cdif 3960  {csn 4631  cmpt 5231  ccnv 5688  cima 5692  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  mptsuppd  8211  extmptsuppeq  8212  suppssov1  8221  suppssov2  8222  suppss2  8224  suppssfv  8226
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