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Theorem mptsuppd 7183
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 7182 . 2 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
5 mptsuppd.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑈)
6 elex 3184 . . . . . 6 (𝐵𝑈𝐵 ∈ V)
75, 6syl 17 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
87biantrurd 527 . . . 4 ((𝜑𝑥𝐴) → (𝐵𝑍 ↔ (𝐵 ∈ V ∧ 𝐵𝑍)))
9 eldifsn 4259 . . . 4 (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵𝑍))
108, 9syl6rbbr 277 . . 3 ((𝜑𝑥𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵𝑍))
1110rabbidva 3162 . 2 (𝜑 → {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})} = {𝑥𝐴𝐵𝑍})
124, 11eqtrd 2643 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵𝑍})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  Vcvv 3172  cdif 3536  {csn 4124  cmpt 4637  (class class class)co 6527   supp csupp 7160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-supp 7161
This theorem is referenced by:  rmsupp0  41965  domnmsuppn0  41966  rmsuppss  41967  suppmptcfin  41976  lcoc0  42027  linc1  42030
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