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Theorem fmpt2d 7144
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 3144 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
3 eqid 2735 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6709 . . . 4 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 fmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6662 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 257 . 2 (𝜑𝐹 Fn 𝐴)
9 fmpt2d.3 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
109ralrimiva 3144 . 2 (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶)
11 ffnfv 7139 . 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 395   = wceq 1537  wcel 2106  wral 3059  cmpt 5231   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  cantnff  9712  limsupgre  15514  idaf  18117  curfcl  18289  ghmqusnsg  19313  ghmquskerlem3  19317  ghmqusker  19318  mat2pmatf  22750  m2cpmf  22764  pm2mpf  22820  clsf  23072  kgenf  23565  lgamf  27100  vmaf  27177  lgsdchr  27414  mirf  28683  suppovss  32696  omsf  34278  erdszelem6  35181  cdleme50f  40525  dochfN  41339  binomcxplemdvsum  44351
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