Theorem fvmpt2d 7012

Description: Deduction version of fvmpt2 7010. (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 6893	. . 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 2727	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fvmpt2 7010	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
8	4, 5, 7	syl2an2 685	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
9	3, 8	eqtrd 2767	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5225  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  cantnflem1  9704  ghmquskerco  19226  frlmphl  21702  neiptopreu  23024  rrxds  25308  ofoprabco  32433  suppovss  32448  tocycf  32816  ply1moneq  33196  fedgmullem2  33260  esumcvg  33641  ofcfval2  33659  eulerpartgbij  33928  dstrvprob  34027  itgexpif  34174  hgt750lemb  34224  frlmsnic  41692  cvgdvgrat  43673  radcnvrat  43674  binomcxplemnotnn0  43716  fmuldfeqlem1  44893  climreclmpt  44995  climinfmpt  45026  limsupubuzmpt  45030  limsupre2mpt  45041  limsupre3mpt  45045  limsupreuzmpt  45050  liminfvalxrmpt  45097  liminflbuz2  45126  cncficcgt0  45199  dvdivbd  45234  dvnmul  45254  dvnprodlem1  45257  dvnprodlem2  45258  stoweidlem42  45353  dirkeritg  45413  elaa2lem  45544  etransclem4  45549  ioorrnopnxrlem  45617  subsaliuncllem  45668  meaiuninclem  45791  meaiininclem  45797  ovnhoilem1  45912  ovncvr2  45922  ovolval4lem1  45960  iccvonmbllem  45989  vonioolem1  45991  vonioolem2  45992  vonicclem1  45994  vonicclem2  45995  pimconstlt0  46012  pimconstlt1  46013  smfpimltmpt  46057  issmfdmpt  46059  smfaddlem2  46075  smflimlem2  46083  smflimlem4  46085  smfpimgtmpt  46092  smfmullem4  46105  smfpimcclem  46118  smfsuplem1  46122  smfsupmpt  46126  smfinfmpt  46130  smflimsuplem2  46132  smflimsuplem3  46133  smflimsuplem4  46134  fsupdm  46153  finfdm  46157