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Theorem dfmpt3 6350
Description: Alternate definition for the maps-to notation df-mpt 5042. (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 5042 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 velsn 4488 . . . . . . 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 1830 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
6 eliunxp 5594 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})))
7 elopab 5304 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
85, 6, 73bitr4i 304 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
98eqriv 2792 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
101, 9eqtr4i 2822 1 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1522  wex 1761  wcel 2081  {csn 4472  cop 4478   ciun 4825  {copab 5024  cmpt 5041   × cxp 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-iun 4827  df-opab 5025  df-mpt 5042  df-xp 5449  df-rel 5450
This theorem is referenced by:  dfmpt  6769  taylpfval  24636  indval2  30890
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