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Theorem fvmptd2 7006
Description: Deduction version of fvmpt 6998 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fvmptd2.1 𝐹 = (𝑥𝐷𝐵)
fvmptd2.2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
fvmptd2.3 (𝜑𝐴𝐷)
fvmptd2.4 (𝜑𝐶𝑉)
Assertion
Ref Expression
fvmptd2 (𝜑 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd2
StepHypRef Expression
1 fvmptd2.1 . . 3 𝐹 = (𝑥𝐷𝐵)
21a1i 11 . 2 (𝜑𝐹 = (𝑥𝐷𝐵))
3 fvmptd2.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
4 fvmptd2.3 . 2 (𝜑𝐴𝐷)
5 fvmptd2.4 . 2 (𝜑𝐶𝑉)
62, 3, 4, 5fvmptd 7005 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cmpt 5231  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  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-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  fvmptopabOLD  7467  updjudhcoinlf  9930  updjudhcoinrg  9931  lcmf0val  16564  fvprmselelfz  16982  fvprmselgcd1  16983  setcval  18032  catcval  18055  estrcval  18080  hofval  18210  yonval  18219  frmdval  18769  smndex1igid  18822  smndex1n0mnd  18830  gexval  19488  pmatcollpw3fi1lem1  22509  chfacfscmul0  22581  chfacfscmulgsum  22583  chfacfpmmul0  22585  chfacfpmmulgsum  22587  lmfval  22957  kgenval  23260  ptval  23295  utopval  23958  ustuqtoplem  23965  utopsnneiplem  23973  tusval  23991  blfvalps  24110  tmsval  24210  metuval  24279  caufval  25024  dchrval  26974  gausslemma2dlem2  27107  gausslemma2dlem3  27108  israg  28216  perpln1  28229  perpln2  28230  isperp  28231  vtxdgfval  28992  crctcsh  29346  clwlkclwwlklem2fv1  29516  clwlkclwwlklem2fv2  29517  cofmpt2  32126  pwrssmgc  32438  frobrhm  32653  qusima  32794  elrspunidl  32821  elrspunsn  32822  r1pquslmic  32957  metidval  33169  pstmval  33174  carsgval  33601  bj-rdg0gALT  36256  bj-finsumval0  36470  cdleme31fv2  39568  iunrelexpmin1  42762  iunrelexpmin2  42766  rfovcnvf1od  43058  limsup10exlem  44787  prproropf1olem3  46472  prprval  46481  clintopval  46881  rngcval  46949  ringcval  46995  1arymaptfo  47417  2arymptfv  47424  2arymaptfo  47428  ackval42  47470
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