Theorem dfmpt3 5445

Description: Alternate definition for the maps-to notation df-mpt 4146. (Contributed by Mario Carneiro, 30-Dec-2016.)

Assertion

Ref	Expression
dfmpt3	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})

Proof of Theorem dfmpt3

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4146	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		velsn 3683	. . . . . . 7 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
3	2	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
4	3	anbi2i 457	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
5	4	2exbii 1652	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
6		eliunxp 4860	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})))
7		elopab 4345	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)))
8	5, 6, 7	3bitr4i 212	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)})
9	8	eqriv 2226	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10	1, 9	eqtr4i 2253	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {csn 3666  ⟨cop 3669  ∪ ciun 3964  {copab 4143   ↦ cmpt 4144   × cxp 4716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iun 3966  df-opab 4145  df-mpt 4146  df-xp 4724  df-rel 4725

This theorem is referenced by:  dfmpt  5811  dfmptg  5813