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Theorem cnvimadfsn 8156
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

Proof of Theorem cnvimadfsn
StepHypRef Expression
1 dfima3 6062 . 2 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)}
2 eldifvsn 4800 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍))
32elv 3480 . . . . 5 (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍)
4 vex 3478 . . . . . . 7 𝑦 ∈ V
5 vex 3478 . . . . . . 7 𝑥 ∈ V
64, 5opelcnv 5881 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 df-br 5149 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
86, 7bitr4i 277 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅𝑥𝑅𝑦)
93, 8anbi12ci 628 . . . 4 ((𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑍))
109exbii 1850 . . 3 (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
1110abbii 2802 . 2 {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
121, 11eqtri 2760 1 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  Vcvv 3474  cdif 3945  {csn 4628  cop 4634   class class class wbr 5148  ccnv 5675  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  suppimacnvss  8157  suppimacnv  8158
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