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Theorem relintab 39823
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 6043 . . 3 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 4177 . . 3 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2844 . 2 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
4 dfrel2 6040 . . 3 (Rel {𝑥𝜑} ↔ {𝑥𝜑} = {𝑥𝜑})
54biimpi 217 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑥𝜑})
6 relintabex 39821 . . . 4 (Rel {𝑥𝜑} → ∃𝑥𝜑)
76xpinintabd 39820 . . 3 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8 incom 4177 . . . . . . . . 9 ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9 cnvcnv 6043 . . . . . . . . 9 𝑥 = (𝑥 ∩ (V × V))
108, 9eqtr4i 2847 . . . . . . . 8 ((V × V) ∩ 𝑥) = 𝑥
1110eqeq2i 2834 . . . . . . 7 (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = 𝑥)
1211anbi1i 623 . . . . . 6 ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = 𝑥𝜑))
1312exbii 1839 . . . . 5 (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = 𝑥𝜑))
1413rabbii 3474 . . . 4 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
1514inteqi 4873 . . 3 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
167, 15syl6eq 2872 . 2 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
173, 5, 163eqtr3a 2880 1 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  {cab 2799  {crab 3142  Vcvv 3495  cin 3934  𝒫 cpw 4537   cint 4869   × cxp 5547  ccnv 5548  Rel wrel 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-int 4870  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557
This theorem is referenced by: (None)
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