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 Description: The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
cnvimadfsn (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑍,𝑦

StepHypRef Expression
1 dfima3 5926 . 2 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)}
2 eldifvsn 4723 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍))
32elv 3499 . . . . 5 (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍)
4 vex 3497 . . . . . . 7 𝑦 ∈ V
5 vex 3497 . . . . . . 7 𝑥 ∈ V
64, 5opelcnv 5746 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 df-br 5059 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
86, 7bitr4i 280 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅𝑥𝑅𝑦)
93, 8anbi12ci 629 . . . 4 ((𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑍))
109exbii 1844 . . 3 (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
1110abbii 2886 . 2 {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
121, 11eqtri 2844 1 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799   ≠ wne 3016  Vcvv 3494   ∖ cdif 3932  {csn 4560  ⟨cop 4566   class class class wbr 5058  ◡ccnv 5548   “ cima 5552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562 This theorem is referenced by:  suppimacnvss  7834  suppimacnv  7835
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