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Theorem mptsuppd 7556
Description: The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
Hypotheses
Ref Expression
mptsuppdifd.f 𝐹 = (𝑥𝐴𝐵)
mptsuppdifd.a (𝜑𝐴𝑉)
mptsuppdifd.z (𝜑𝑍𝑊)
mptsuppd.b ((𝜑𝑥𝐴) → 𝐵𝑈)
Assertion
Ref Expression
mptsuppd (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵𝑍})
Distinct variable groups:   𝑥,𝐴   𝑥,𝑍   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑈(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptsuppd
StepHypRef Expression
1 mptsuppdifd.f . . 3 𝐹 = (𝑥𝐴𝐵)
2 mptsuppdifd.a . . 3 (𝜑𝐴𝑉)
3 mptsuppdifd.z . . 3 (𝜑𝑍𝑊)
41, 2, 3mptsuppdifd 7555 . 2 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
5 mptsuppd.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑈)
6 elex 3401 . . . . . 6 (𝐵𝑈𝐵 ∈ V)
75, 6syl 17 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
87biantrurd 529 . . . 4 ((𝜑𝑥𝐴) → (𝐵𝑍 ↔ (𝐵 ∈ V ∧ 𝐵𝑍)))
9 eldifsn 4507 . . . 4 (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵𝑍))
108, 9syl6rbbr 282 . . 3 ((𝜑𝑥𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵𝑍))
1110rabbidva 3373 . 2 (𝜑 → {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})} = {𝑥𝐴𝐵𝑍})
124, 11eqtrd 2834 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2972  {crab 3094  Vcvv 3386  cdif 3767  {csn 4369  cmpt 4923  (class class class)co 6879   supp csupp 7533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-supp 7534
This theorem is referenced by:  rmsupp0  42943  domnmsuppn0  42944  rmsuppss  42945  suppmptcfin  42954  lcoc0  43005  linc1  43008
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