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Theorem cnvimadfsn 7349
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 5504 . 2 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)}
2 vex 3234 . . . . . 6 𝑦 ∈ V
3 eldifvsn 4359 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍))
42, 3ax-mp 5 . . . . 5 (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍)
5 vex 3234 . . . . . . 7 𝑥 ∈ V
62, 5opelcnv 5336 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 df-br 4686 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
86, 7bitr4i 267 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅𝑥𝑅𝑦)
94, 8anbi12ci 734 . . . 4 ((𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑍))
109exbii 1814 . . 3 (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
1110abbii 2768 . 2 {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
121, 11eqtri 2673 1 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  Vcvv 3231  cdif 3604  {csn 4210  cop 4216   class class class wbr 4685  ccnv 5142  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  suppimacnvss  7350  suppimacnv  7351
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