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Theorem mptsuppdifd 8212
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 7245 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ V)
41, 3eqeltrid 2844 . . 3 (𝜑𝐹 ∈ V)
5 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
6 suppimacnv 8200 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 584 . 2 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
81mptpreima 6257 . 2 (𝐹 “ (V ∖ {𝑍})) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})}
97, 8eqtrdi 2792 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  cdif 3947  {csn 4625  cmpt 5224  ccnv 5683  cima 5687  (class class class)co 7432   supp csupp 8186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-supp 8187
This theorem is referenced by:  mptsuppd  8213  extmptsuppeq  8214  suppssov1  8223  suppssov2  8224  suppss2  8226  suppssfv  8228
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