Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptsuppd Structured version   Visualization version   GIF version

Theorem mptsuppd 7488
 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 7487 . 2 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})})
5 mptsuppd.b . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑈)
6 elex 3353 . . . . . 6 (𝐵𝑈𝐵 ∈ V)
75, 6syl 17 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
87biantrurd 530 . . . 4 ((𝜑𝑥𝐴) → (𝐵𝑍 ↔ (𝐵 ∈ V ∧ 𝐵𝑍)))
9 eldifsn 4463 . . . 4 (𝐵 ∈ (V ∖ {𝑍}) ↔ (𝐵 ∈ V ∧ 𝐵𝑍))
108, 9syl6rbbr 279 . . 3 ((𝜑𝑥𝐴) → (𝐵 ∈ (V ∖ {𝑍}) ↔ 𝐵𝑍))
1110rabbidva 3329 . 2 (𝜑 → {𝑥𝐴𝐵 ∈ (V ∖ {𝑍})} = {𝑥𝐴𝐵𝑍})
124, 11eqtrd 2795 1 (𝜑 → (𝐹 supp 𝑍) = {𝑥𝐴𝐵𝑍})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2140   ≠ wne 2933  {crab 3055  Vcvv 3341   ∖ cdif 3713  {csn 4322   ↦ cmpt 4882  (class class class)co 6815   supp csupp 7465 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-supp 7466 This theorem is referenced by:  rmsupp0  42678  domnmsuppn0  42679  rmsuppss  42680  suppmptcfin  42689  lcoc0  42740  linc1  42743
 Copyright terms: Public domain W3C validator