Theorem fvmpt2d 7029

Description: Deduction version of fvmpt2 7027. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Hypotheses

Ref	Expression
fvmpt2d.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
fvmpt2d.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
fvmpt2d	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpt2d

Step	Hyp	Ref	Expression
1		fvmpt2d.1	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
2	1	fveq1d 6908	. . 3 ⊢ (𝜑 → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
4		id 22	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
5		fvmpt2d.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
6		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fvmpt2 7027	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
8	4, 5, 7	syl2an2 686	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	3, 8	eqtrd 2777	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5225  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569

This theorem is referenced by:  cantnflem1  9729  ghmquskerco  19302  frlmphl  21801  neiptopreu  23141  rrxds  25427  ofoprabco  32674  suppovss  32690  tocycf  33137  elrgspnsubrunlem2  33252  ply1moneq  33611  fedgmullem2  33681  esumcvg  34087  ofcfval2  34105  eulerpartgbij  34374  dstrvprob  34474  itgexpif  34621  hgt750lemb  34671  aks6d1c6lem4  42174  frlmsnic  42550  cvgdvgrat  44332  radcnvrat  44333  binomcxplemnotnn0  44375  fmuldfeqlem1  45597  climreclmpt  45699  climinfmpt  45730  limsupubuzmpt  45734  limsupre2mpt  45745  limsupre3mpt  45749  limsupreuzmpt  45754  liminfvalxrmpt  45801  liminflbuz2  45830  cncficcgt0  45903  dvdivbd  45938  dvnmul  45958  dvnprodlem1  45961  dvnprodlem2  45962  stoweidlem42  46057  dirkeritg  46117  elaa2lem  46248  etransclem4  46253  ioorrnopnxrlem  46321  subsaliuncllem  46372  meaiuninclem  46495  meaiininclem  46501  ovnhoilem1  46616  ovncvr2  46626  ovolval4lem1  46664  iccvonmbllem  46693  vonioolem1  46695  vonioolem2  46696  vonicclem1  46698  vonicclem2  46699  pimconstlt0  46716  pimconstlt1  46717  smfpimltmpt  46761  issmfdmpt  46763  smfaddlem2  46779  smflimlem2  46787  smflimlem4  46789  smfpimgtmpt  46796  smfmullem4  46809  smfpimcclem  46822  smfsuplem1  46826  smfsupmpt  46830  smfinfmpt  46834  smflimsuplem2  46836  smflimsuplem3  46837  smflimsuplem4  46838  fsupdm  46857  finfdm  46861  tposcurf1  48999  fucocolem4  49051