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Theorem dfmpt3 6175
Description: Alternate definition for the "maps to" notation df-mpt 4882. (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 4882 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 velsn 4337 . . . . . . 7 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
32anbi2i 732 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
43anbi2i 732 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
542exbii 1924 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
6 eliunxp 5415 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})))
7 elopab 5133 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
85, 6, 73bitr4i 292 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
98eqriv 2757 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
101, 9eqtr4i 2785 1 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {csn 4321  cop 4327   ciun 4672  {copab 4864  cmpt 4881   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273
This theorem is referenced by:  dfmpt  6574  taylpfval  24338  indval2  30406
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