Theorem fnasrn 6960

Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfmpt.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fnasrn	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)

Proof of Theorem fnasrn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmpt.1	. . 3 ⊢ 𝐵 ∈ V
2	1	dfmpt 6959	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
3		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
4	3	rnmpt 5824	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5		velsn 4557	. . . . . 6 ⊢ (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
6	5	rexbii 3170	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
7	6	abbii 2808	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
8	4, 7	eqtr4i 2768	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9		df-iun 4906	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
10	8, 9	eqtr4i 2768	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
11	2, 10	eqtr4i 2768	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)

Syntax hints:   = wceq 1543   ∈ wcel 2110  {cab 2714  ∃wrex 3062  Vcvv 3408  {csn 4541  ⟨cop 4547  ∪ ciun 4904   ↦ cmpt 5135  ran crn 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387

This theorem is referenced by:  idref  6961  resfunexg  7031  gruf  10425