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Theorem cnvimadfsn 7264
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 5438 . 2 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)}
2 vex 3193 . . . . . 6 𝑦 ∈ V
3 eldifvsn 4302 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍))
42, 3ax-mp 5 . . . . 5 (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦𝑍)
5 vex 3193 . . . . . . 7 𝑥 ∈ V
62, 5opelcnv 5274 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 df-br 4624 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
86, 7bitr4i 267 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝑅𝑥𝑅𝑦)
94, 8anbi12ci 733 . . . 4 ((𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑍))
109exbii 1771 . . 3 (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
1110abbii 2736 . 2 {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
121, 11eqtri 2643 1 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  Vcvv 3190  cdif 3557  {csn 4155  cop 4161   class class class wbr 4623  ccnv 5083  cima 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by:  suppimacnvss  7265  suppimacnv  7266
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