Theorem fvmptd2 6770

Description: Deduction version of fvmpt 6762 (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 = (𝑥 ∈ 𝐷 ↦ 𝐵)
fvmptd2.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
fvmptd2.3	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fvmptd2.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fvmptd2	⊢ (𝜑 → (𝐹‘𝐴) = 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

Step	Hyp	Ref	Expression
1		fvmptd2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
3		fvmptd2.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
4		fvmptd2.3	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5		fvmptd2.4	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
6	2, 3, 4, 5	fvmptd 6769	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ↦ cmpt 5138  ‘cfv 6349

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357

This theorem is referenced by:  fvmptopab  7203  updjudhcoinlf  9355  updjudhcoinrg  9356  lcmf0val  15960  fvprmselelfz  16374  fvprmselgcd1  16375  setcval  17331  catcval  17350  estrcval  17368  hofval  17496  yonval  17505  frmdval  18010  smndex1igid  18063  smndex1n0mnd  18071  gexval  18697  pmatcollpw3fi1lem1  21388  chfacfscmul0  21460  chfacfscmulgsum  21462  chfacfpmmul0  21464  chfacfpmmulgsum  21466  lmfval  21834  kgenval  22137  ptval  22172  utopval  22835  ustuqtoplem  22842  utopsnneiplem  22850  tusval  22869  blfvalps  22987  tmsval  23085  metuval  23153  caufval  23872  dchrval  25804  gausslemma2dlem2  25937  gausslemma2dlem3  25938  israg  26477  perpln1  26490  perpln2  26491  isperp  26492  vtxdgfval  27243  crctcsh  27596  clwlkclwwlklem2fv1  27767  clwlkclwwlklem2fv2  27768  cofmpt2  30373  carsgval  31556  bj-finsumval0  34561  cdleme31fv2  37523  iunrelexpmin1  40046  iunrelexpmin2  40050  rfovcnvf1od  40343  limsup10exlem  42046  prproropf1olem3  43661  prprval  43670  clintopval  44105  rngcval  44227  ringcval  44273