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Theorem fmpt2d 7123
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 3147 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
3 eqid 2733 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 6691 . . . 4 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 fmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 6643 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 257 . 2 (𝜑𝐹 Fn 𝐴)
9 fmpt2d.3 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
109ralrimiva 3147 . 2 (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶)
11 ffnfv 7118 . 2 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶))
128, 10, 11sylanbrc 584 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  cmpt 5232   Fn wfn 6539  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  cantnff  9669  limsupgre  15425  idaf  18013  curfcl  18185  mat2pmatf  22230  m2cpmf  22244  pm2mpf  22300  clsf  22552  kgenf  23045  lgamf  26546  vmaf  26623  lgsdchr  26858  mirf  27942  suppovss  31937  ghmquskerlem3  32562  ghmqusker  32563  omsf  33326  erdszelem6  34218  cdleme50f  39461  dochfN  40275  binomcxplemdvsum  43162
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