Theorem mpoeq3ia 7511

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 1129	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
3	2	mpoeq3dva 7510	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
4	3	mptru 1544	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ⊤wtru 1538   ∈ wcel 2106   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  mpodifsnif  7548  mposnif  7549  oprab2co  8121  cnfcomlem  9737  cnfcom2  9740  dfioo2  13487  elovmpowrd  14593  sadcom  16497  comfffval2  17746  oppchomf  17767  symgga  19440  oppglsm  19675  dfrhm2  20491  cnfldsub  21428  cnflddiv  21431  cnflddivOLD  21432  mat0op  22441  mattpos1  22478  mdetunilem7  22640  madufval  22659  maducoeval2  22662  madugsum  22665  mp2pm2mplem5  22832  mp2pm2mp  22833  leordtval  23237  xpstopnlem1  23833  divcnOLD  24904  divcn  24906  oprpiece1res1  24996  oprpiece1res2  24997  ehl1eudis  25468  ehl2eudis  25470  cxpcn  26802  cxpcnOLD  26803  cnnvm  30711  mdetpmtr2  33785  madjusmdetlem1  33788  cnre2csqima  33872  mndpluscn  33887  raddcn  33890  icorempo  37334  matunitlindflem1  37603  mendplusgfval  43170  hoidmv1le  46550  hspdifhsp  46572  vonn0ioo  46643  vonn0icc  46644  dflinc2  48256