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Theorem dfmpt3 5101
Description: Alternate definition for the maps-to notation df-mpt 3876. (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 3876 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 velsn 3448 . . . . . . 7 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
32anbi2i 445 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
43anbi2i 445 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
542exbii 1540 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
6 eliunxp 4543 . . . 4 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 ∈ {𝐵})))
7 elopab 4059 . . . 4 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦 = 𝐵)))
85, 6, 73bitr4i 210 . . 3 (𝑧 𝑥𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
98eqriv 2082 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
101, 9eqtr4i 2108 1 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1287  wex 1424  wcel 1436  {csn 3431  cop 3434   ciun 3713  {copab 3873  cmpt 3874   × cxp 4409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3715  df-opab 3875  df-mpt 3876  df-xp 4417  df-rel 4418
This theorem is referenced by:  dfmpt  5437  dfmptg  5439
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