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Theorem fnasrn 7154
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 7153 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
3 eqid 2728 . . . . 5 (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
43rnmpt 5957 . . . 4 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5 velsn 4645 . . . . . 6 (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
65rexbii 3091 . . . . 5 (∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
76abbii 2798 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
84, 7eqtr4i 2759 . . 3 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9 df-iun 4998 . . 3 𝑥𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
108, 9eqtr4i 2759 . 2 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
112, 10eqtr4i 2759 1 (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  Vcvv 3471  {csn 4629  cop 4635   ciun 4996  cmpt 5231  ran crn 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555
This theorem is referenced by:  idref  7155  resfunexg  7227  gruf  10835
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