Theorem dfmpt 7086

Description: Alternate definition for the maps-to notation df-mpt 5177 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Hypothesis

Ref	Expression
dfmpt.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfmpt	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}

Proof of Theorem dfmpt

Step	Hyp	Ref	Expression
1		dfmpt3 6623	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})
2		vex 3441	. . . . 5 ⊢ 𝑥 ∈ V
3		dfmpt.1	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	xpsn 7083	. . . 4 ⊢ ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
6	5	iuneq2i 4965	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
7	1, 6	eqtri 2756	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4577  ⟨cop 4583  ∪ ciun 4943   ↦ cmpt 5176   × cxp 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by:  fnasrn  7087  funiun  7089  dfmpo  8041