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Theorem relintab 36908
 Description: Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)
Assertion
Ref Expression
relintab (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
Distinct variable groups:   𝜑,𝑤   𝑥,𝑤
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem relintab
StepHypRef Expression
1 cnvcnv 5505 . . 3 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 3767 . . 3 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2632 . 2 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
4 dfrel2 5502 . . 3 (Rel {𝑥𝜑} ↔ {𝑥𝜑} = {𝑥𝜑})
54biimpi 205 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑥𝜑})
6 relintabex 36906 . . . 4 (Rel {𝑥𝜑} → ∃𝑥𝜑)
76xpinintabd 36905 . . 3 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8 incom 3767 . . . . . . . . . 10 ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9 cnvcnv 5505 . . . . . . . . . 10 𝑥 = (𝑥 ∩ (V × V))
108, 9eqtr4i 2635 . . . . . . . . 9 ((V × V) ∩ 𝑥) = 𝑥
1110eqeq2i 2622 . . . . . . . 8 (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = 𝑥)
1211anbi1i 727 . . . . . . 7 ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = 𝑥𝜑))
1312exbii 1764 . . . . . 6 (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = 𝑥𝜑))
1413a1i 11 . . . . 5 (𝑤 ∈ 𝒫 (V × V) → (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = 𝑥𝜑)))
1514rabbiia 3161 . . . 4 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
1615inteqi 4414 . . 3 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
177, 16syl6eq 2660 . 2 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
183, 5, 173eqtr3a 2668 1 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∩ cin 3539  𝒫 cpw 4108  ∩ cint 4410   × cxp 5036  ◡ccnv 5037  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-int 4411  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by: (None)
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