Theorem fvmpt2d 6293

Description: Deduction version of fvmpt2 6291. (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 6193	. . 3 ⊢ (𝜑 → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
3	2	adantr 481	. 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 2622	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fvmpt2 6291	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
8	4, 5, 7	syl2an2 875	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	3, 8	eqtrd 2656	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  cantnflem1  8586  frlmphl  20120  neiptopreu  20937  rrxds  23181  ofoprabco  29464  esumcvg  30148  ofcfval2  30166  eulerpartgbij  30434  dstrvprob  30533  itgexpif  30684  hgt750lemb  30734  cvgdvgrat  38512  radcnvrat  38513  binomcxplemnotnn0  38555  fmuldfeqlem1  39814  climreclmpt  39916  climinfmpt  39947  limsupubuzmpt  39951  limsupre2mpt  39962  limsupre3mpt  39966  limsupreuzmpt  39971  liminfvalxrmpt  40018  cncficcgt0  40101  dvdivbd  40138  dvnmul  40158  dvnprodlem1  40161  dvnprodlem2  40162  stoweidlem42  40259  dirkeritg  40319  elaa2lem  40450  etransclem4  40455  ioorrnopnxrlem  40526  subsaliuncllem  40575  meaiuninclem  40697  meaiininclem  40700  ovnhoilem1  40815  ovncvr2  40825  ovolval4lem1  40863  iccvonmbllem  40892  vonioolem1  40894  vonioolem2  40895  vonicclem1  40897  vonicclem2  40898  pimconstlt0  40914  pimconstlt1  40915  pimgtmnf  40932  smfpimltmpt  40955  issmfdmpt  40957  smfpimltxrmpt  40967  smfaddlem2  40972  smflimlem2  40980  smflimlem4  40982  smfpimgtmpt  40989  smfpimgtxrmpt  40992  smfmullem4  41001  smfpimcclem  41013  smfsuplem1  41017  smfsupmpt  41021  smfinfmpt  41025  smflimsuplem2  41027  smflimsuplem3  41028  smflimsuplem4  41029