Theorem dfmpt 7097

Description: Alternate definition for the maps-to notation df-mpt 5167 (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 6632	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})
2		vex 3433	. . . . 5 ⊢ 𝑥 ∈ V
3		dfmpt.1	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	xpsn 7094	. . . 4 ⊢ ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
6	5	iuneq2i 4955	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
7	1, 6	eqtri 2759	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ⟨cop 4573  ∪ ciun 4933   ↦ cmpt 5166   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505

This theorem is referenced by:  fnasrn  7098  funiun  7100  dfmpo  8052