Theorem fvmptd2 6596

Description: Deduction version of fvmpt 6589 (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 6595	1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048   ↦ cmpt 5002  ‘cfv 6182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190

This theorem is referenced by:  fvmptopab  7022  updjudhcoinlf  9147  updjudhcoinrg  9148  lcmf0val  15812  fvprmselelfz  16226  fvprmselgcd1  16227  setcval  17185  catcval  17204  estrcval  17222  hofval  17350  yonval  17359  frmdval  17847  gexval  18454  pmatcollpw3fi1lem1  21088  chfacfscmul0  21160  chfacfscmulgsum  21162  chfacfpmmul0  21164  chfacfpmmulgsum  21166  lmfval  21534  kgenval  21837  ptval  21872  utopval  22534  ustuqtoplem  22541  utopsnneiplem  22549  tusval  22568  blfvalps  22686  tmsval  22784  metuval  22852  caufval  23571  dchrval  25502  gausslemma2dlem2  25635  gausslemma2dlem3  25636  israg  26175  perpln1  26188  perpln2  26189  isperp  26190  vtxdgfval  26942  crctcsh  27300  clwlkclwwlklem2fv1  27491  clwlkclwwlklem2fv2  27492  cofmpt2  30131  carsgval  31163  cdleme31fv2  36922  limsup10exlem  41430  prproropf1olem3  42975  prprval  42984  clintopval  43415  rngcval  43537  ringcval  43583