Theorem mpoeq3ia 7489

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpoeq3ia.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
mpoeq3ia	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Proof of Theorem mpoeq3ia

Step	Hyp	Ref	Expression
1		mpoeq3ia.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
2	1	3adant1 1130	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
3	2	mpoeq3dva 7488	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
4	3	mptru 1548	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106   ∈ cmpo 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  mpodifsnif  7525  mposnif  7526  oprab2co  8085  cnfcomlem  9696  cnfcom2  9699  dfioo2  13431  elovmpowrd  14512  sadcom  16408  comfffval2  17649  oppchomf  17670  symgga  19316  oppglsm  19551  dfrhm2  20365  cnfldsub  21173  cnflddiv  21175  mat0op  22141  mattpos1  22178  mdetunilem7  22340  madufval  22359  maducoeval2  22362  madugsum  22365  mp2pm2mplem5  22532  mp2pm2mp  22533  leordtval  22937  xpstopnlem1  23533  divcnOLD  24604  divcn  24606  oprpiece1res1  24691  oprpiece1res2  24692  ehl1eudis  25161  ehl2eudis  25163  cxpcn  26477  cnnvm  30190  mdetpmtr2  33090  madjusmdetlem1  33093  cnre2csqima  33177  mndpluscn  33192  raddcn  33195  gg-cxpcn  35470  icorempo  36535  matunitlindflem1  36787  mendplusgfval  42229  hoidmv1le  45609  hspdifhsp  45631  vonn0ioo  45702  vonn0icc  45703  dflinc2  47179