MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmpt2d Structured version   Visualization version   GIF version

Theorem fmpt2d 7125
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmpt2d.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
fmpt2d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fmpt2d.3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
Assertion
Ref Expression
fmpt2d (𝜑𝐹:𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐶   𝑦,𝐹   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem fmpt2d
StepHypRef Expression
1 fmpt2d.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
21ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
3 eqid 2732 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6690 . . . 4 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 fmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6642 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 256 . 2 (𝜑𝐹 Fn 𝐴)
9 fmpt2d.3 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
109ralrimiva 3146 . 2 (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶)
11 ffnfv 7120 . 2 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶))
128, 10, 11sylanbrc 583 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cmpt 5231   Fn wfn 6538  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  cantnff  9671  limsupgre  15429  idaf  18017  curfcl  18189  mat2pmatf  22450  m2cpmf  22464  pm2mpf  22520  clsf  22772  kgenf  23265  lgamf  26770  vmaf  26847  lgsdchr  27082  mirf  28166  suppovss  32161  ghmquskerlem3  32793  ghmqusker  32794  omsf  33581  erdszelem6  34473  cdleme50f  39716  dochfN  40530  binomcxplemdvsum  43416
  Copyright terms: Public domain W3C validator