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Theorem mptsuppdifd 7846
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 6985 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ V)
41, 3eqeltrid 2921 . . 3 (𝜑𝐹 ∈ V)
5 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
6 suppimacnv 7835 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 584 . 2 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
81mptpreima 6089 . 2 (𝐹 “ (V ∖ {𝑍})) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})}
97, 8syl6eq 2876 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2106  {crab 3146  Vcvv 3499  cdif 3936  {csn 4563  cmpt 5142  ccnv 5552  cima 5556  (class class class)co 7151   supp csupp 7824
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-supp 7825
This theorem is referenced by:  mptsuppd  7847  extmptsuppeq  7848  suppssov1  7856  suppss2  7858  suppssfv  7860
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