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 1128	. . 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 1546	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539  ⊤wtru 1540   ∈ wcel 2104   ∈ cmpo 7413

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 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  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  24696  oprpiece1res2  24697  ehl1eudis  25168  ehl2eudis  25170  cxpcn  26489  cnnvm  30202  mdetpmtr2  33102  madjusmdetlem1  33105  cnre2csqima  33189  mndpluscn  33204  raddcn  33207  gg-cxpcn  35470  icorempo  36535  matunitlindflem1  36787  mendplusgfval  42229  hoidmv1le  45608  hspdifhsp  45630  vonn0ioo  45701  vonn0icc  45702  dflinc2  47178