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Theorem fnasrn 6298
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 6297 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
3 eqid 2605 . . . . 5 (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
43rnmpt 5275 . . . 4 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5 velsn 4136 . . . . . 6 (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
65rexbii 3018 . . . . 5 (∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
76abbii 2721 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
84, 7eqtr4i 2630 . . 3 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9 df-iun 4447 . . 3 𝑥𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
108, 9eqtr4i 2630 . 2 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
112, 10eqtr4i 2630 1 (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1975  {cab 2591  wrex 2892  Vcvv 3168  {csn 4120  cop 4126   ciun 4445  cmpt 4633  ran crn 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793
This theorem is referenced by:  resfunexg  6358  idref  6377  gruf  9485
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