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Theorem fnasrn 7074
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 (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)

Proof of Theorem fnasrn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfmpt.1 . . 3 𝐵 ∈ V
21dfmpt 7073 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
3 eqid 2736 . . . . 5 (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
43rnmpt 5897 . . . 4 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5 velsn 4590 . . . . . 6 (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
65rexbii 3093 . . . . 5 (∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
76abbii 2806 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
84, 7eqtr4i 2767 . . 3 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9 df-iun 4944 . . 3 𝑥𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
108, 9eqtr4i 2767 . 2 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
112, 10eqtr4i 2767 1 (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {cab 2713  wrex 3070  Vcvv 3441  {csn 4574  cop 4580   ciun 4942  cmpt 5176  ran crn 5622
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487
This theorem is referenced by:  idref  7075  resfunexg  7148  gruf  10669
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