Theorem mpteq2da 5183

Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)

Hypotheses

Ref	Expression
mpteq2da.1	⊢ Ⅎ𝑥𝜑
mpteq2da.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
mpteq2da	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Proof of Theorem mpteq2da

Step	Hyp	Ref	Expression
1		mpteq2da.1	. 2 ⊢ Ⅎ𝑥𝜑
2		eqidd 2732	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
3		mpteq2da.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
4	1, 2, 3	mpteq12da 5174	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111   ↦ cmpt 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-opab 5154  df-mpt 5173

This theorem is referenced by:  offval2f  7625  prodeq1f  15810  prodeq2ii  15815  gsumsnfd  19861  gsummoncoe1  22221  cayleyhamilton1  22805  xkocnv  23727  utopsnneiplem  24160  itgeq1f  25697  fpwrelmap  32711  gsummpt2d  33024  elrgspnsubrunlem2  33210  elrspunidl  33388  fedgmullem2  33638  esumf1o  34058  esum2d  34101  itg2addnclem  37710  ftc1anclem5  37736  mzpsubmpt  42775  mzpexpmpt  42777  refsum2cnlem1  45073  mpteq2dfa  45303  fmuldfeqlem1  45621  limsupval3  45729  liminfval5  45802  liminfvalxrmpt  45823  liminfval4  45826  limsupval4  45831  liminfvaluz2  45832  limsupvaluz4  45837  cncfiooicclem1  45930  dvmptfprodlem  45981  stoweidlem2  46039  stoweidlem6  46043  stoweidlem8  46045  stoweidlem17  46054  stoweidlem19  46056  stoweidlem20  46057  stoweidlem21  46058  stoweidlem22  46059  stoweidlem23  46060  stoweidlem32  46069  stoweidlem36  46073  stoweidlem40  46077  stoweidlem41  46078  stoweidlem47  46084  stirlinglem15  46125  sge0ss  46449  sge0xp  46466  omeiunlempt  46557  hoicvrrex  46593  ovnlecvr2  46647  smfdiv  46834  smfneg  46840  smflimmpt  46847  smfsupmpt  46852  smfinfmpt  46856  smflimsuplem4  46860  smflimsuplem5  46861  smflimsupmpt  46866  smfliminf  46868  smfliminfmpt  46869