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Theorem relintab 42636
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 6190 . . 3 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 4200 . . 3 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2758 . 2 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
4 dfrel2 6187 . . 3 (Rel {𝑥𝜑} ↔ {𝑥𝜑} = {𝑥𝜑})
54biimpi 215 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑥𝜑})
6 relintabex 42634 . . . 4 (Rel {𝑥𝜑} → ∃𝑥𝜑)
76xpinintabd 42633 . . 3 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8 incom 4200 . . . . . . . . 9 ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9 cnvcnv 6190 . . . . . . . . 9 𝑥 = (𝑥 ∩ (V × V))
108, 9eqtr4i 2761 . . . . . . . 8 ((V × V) ∩ 𝑥) = 𝑥
1110eqeq2i 2743 . . . . . . 7 (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = 𝑥)
1211anbi1i 622 . . . . . 6 ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = 𝑥𝜑))
1312exbii 1848 . . . . 5 (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = 𝑥𝜑))
1413rabbii 3436 . . . 4 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
1514inteqi 4953 . . 3 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
167, 15eqtrdi 2786 . 2 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
173, 5, 163eqtr3a 2794 1 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wex 1779  {cab 2707  {crab 3430  Vcvv 3472  cin 3946  𝒫 cpw 4601   cint 4949   × cxp 5673  ccnv 5674  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-int 4950  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683
This theorem is referenced by: (None)
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