Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptd2 Structured version   Visualization version   GIF version

Theorem fvmptd2 6763
 Description: Deduction version of fvmpt 6755 (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 6762 1 (𝜑 → (𝐹𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5114  ‘cfv 6332 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340 This theorem is referenced by:  fvmptopab  7198  updjudhcoinlf  9363  updjudhcoinrg  9364  lcmf0val  15976  fvprmselelfz  16390  fvprmselgcd1  16391  setcval  17349  catcval  17368  estrcval  17386  hofval  17514  yonval  17523  frmdval  18028  smndex1igid  18081  smndex1n0mnd  18089  gexval  18716  pmatcollpw3fi1lem1  21432  chfacfscmul0  21504  chfacfscmulgsum  21506  chfacfpmmul0  21508  chfacfpmmulgsum  21510  lmfval  21878  kgenval  22181  ptval  22216  utopval  22879  ustuqtoplem  22886  utopsnneiplem  22894  tusval  22913  blfvalps  23031  tmsval  23129  metuval  23197  caufval  23920  dchrval  25862  gausslemma2dlem2  25995  gausslemma2dlem3  25996  israg  26535  perpln1  26548  perpln2  26549  isperp  26550  vtxdgfval  27301  crctcsh  27654  clwlkclwwlklem2fv1  27824  clwlkclwwlklem2fv2  27825  cofmpt2  30437  pwrssmgc  30752  frobrhm  30959  qusima  31063  elrspunidl  31075  carsgval  31737  bj-finsumval0  34851  cdleme31fv2  37840  iunrelexpmin1  40580  iunrelexpmin2  40584  rfovcnvf1od  40876  limsup10exlem  42582  prproropf1olem3  44190  prprval  44199  clintopval  44632  rngcval  44754  ringcval  44800  1arymaptfo  45223  2arymptfv  45230  2arymaptfo  45234  ackval42  45276
 Copyright terms: Public domain W3C validator