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Theorem mptsuppdifd 7302
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)
3 mptexg 6469 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
42, 3syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ V)
51, 4syl5eqel 2703 . . 3 (𝜑𝐹 ∈ V)
6 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
7 suppimacnv 7291 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
85, 6, 7syl2anc 692 . 2 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
91mptpreima 5616 . 2 (𝐹 “ (V ∖ {𝑍})) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})}
108, 9syl6eq 2670 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cdif 3564  {csn 4168  cmpt 4720  ccnv 5103  cima 5107  (class class class)co 6635   supp csupp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-supp 7281
This theorem is referenced by:  mptsuppd  7303  extmptsuppeq  7304  suppssov1  7312  suppss2  7314  suppssfv  7316
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