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Theorem relintab 38391
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 5744 . . 3 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 3948 . . 3 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2782 . 2 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
4 dfrel2 5741 . . 3 (Rel {𝑥𝜑} ↔ {𝑥𝜑} = {𝑥𝜑})
54biimpi 206 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑥𝜑})
6 relintabex 38389 . . . 4 (Rel {𝑥𝜑} → ∃𝑥𝜑)
76xpinintabd 38388 . . 3 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8 incom 3948 . . . . . . . . 9 ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9 cnvcnv 5744 . . . . . . . . 9 𝑥 = (𝑥 ∩ (V × V))
108, 9eqtr4i 2785 . . . . . . . 8 ((V × V) ∩ 𝑥) = 𝑥
1110eqeq2i 2772 . . . . . . 7 (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = 𝑥)
1211anbi1i 733 . . . . . 6 ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = 𝑥𝜑))
1312exbii 1923 . . . . 5 (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = 𝑥𝜑))
1413rabbii 3325 . . . 4 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
1514inteqi 4631 . . 3 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
167, 15syl6eq 2810 . 2 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
173, 5, 163eqtr3a 2818 1 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wex 1853  {cab 2746  {crab 3054  Vcvv 3340  cin 3714  𝒫 cpw 4302   cint 4627   × cxp 5264  ccnv 5265  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-int 4628  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274
This theorem is referenced by: (None)
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