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Theorem fnasrn 5393
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 5392 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
3 eqid 2083 . . . . 5 (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
43rnmpt 4630 . . . 4 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5 velsn 3433 . . . . . 6 (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
65rexbii 2378 . . . . 5 (∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
76abbii 2198 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
84, 7eqtr4i 2106 . . 3 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9 df-iun 3700 . . 3 𝑥𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
108, 9eqtr4i 2106 . 2 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
112, 10eqtr4i 2106 1 (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  {cab 2069  wrex 2354  Vcvv 2610  {csn 3416  cop 3419   ciun 3698  cmpt 3859  ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959
This theorem is referenced by:  idref  5448
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