Theorem mpoeq3ia 7483

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 7482	. 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 7407

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 7409  df-mpo 7410

This theorem is referenced by:  mpodifsnif  7519  mposnif  7520  oprab2co  8079  cnfcomlem  9690  cnfcom2  9693  dfioo2  13423  elovmpowrd  14504  sadcom  16400  comfffval2  17641  oppchomf  17662  symgga  19269  oppglsm  19504  dfrhm2  20245  cnfldsub  20965  cnflddiv  20967  mat0op  21912  mattpos1  21949  mdetunilem7  22111  madufval  22130  maducoeval2  22133  madugsum  22136  mp2pm2mplem5  22303  mp2pm2mp  22304  leordtval  22708  xpstopnlem1  23304  divcn  24375  oprpiece1res1  24458  oprpiece1res2  24459  ehl1eudis  24928  ehl2eudis  24930  cxpcn  26242  cnnvm  29922  mdetpmtr2  32792  madjusmdetlem1  32795  cnre2csqima  32879  mndpluscn  32894  raddcn  32897  gg-divcn  35151  gg-cxpcn  35172  icorempo  36220  matunitlindflem1  36472  mendplusgfval  41912  hoidmv1le  45296  hspdifhsp  45318  vonn0ioo  45389  vonn0icc  45390  dflinc2  47044