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Theorem dfmpt3 6689
Description: Alternate definition for the maps-to notation df-mpt 5232. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
dfmpt3 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})

Proof of Theorem dfmpt3
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 5232 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 velsn 4645 . . . . . . 7 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
32anbi2i 622 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
43anbi2i 622 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
542exbii 1844 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
6 eliunxp 5840 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})))
7 elopab 5529 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
85, 6, 73bitr4i 303 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
98eqriv 2725 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
101, 9eqtr4i 2759 1 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wex 1774  wcel 2099  {csn 4629  cop 4635   ciun 4996  {copab 5210  cmpt 5231   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 4998  df-opab 5211  df-mpt 5232  df-xp 5684  df-rel 5685
This theorem is referenced by:  dfmpt  7153  taylpfval  26312  indval2  33633
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