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Theorem relintab 37397
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 5550 . . 3 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 3788 . . 3 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2643 . 2 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
4 dfrel2 5547 . . 3 (Rel {𝑥𝜑} ↔ {𝑥𝜑} = {𝑥𝜑})
54biimpi 206 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑥𝜑})
6 relintabex 37395 . . . 4 (Rel {𝑥𝜑} → ∃𝑥𝜑)
76xpinintabd 37394 . . 3 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8 incom 3788 . . . . . . . . . 10 ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9 cnvcnv 5550 . . . . . . . . . 10 𝑥 = (𝑥 ∩ (V × V))
108, 9eqtr4i 2646 . . . . . . . . 9 ((V × V) ∩ 𝑥) = 𝑥
1110eqeq2i 2633 . . . . . . . 8 (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = 𝑥)
1211anbi1i 730 . . . . . . 7 ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = 𝑥𝜑))
1312exbii 1771 . . . . . 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 3176 . . . 4 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
1615inteqi 4449 . . 3 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)}
177, 16syl6eq 2671 . 2 (Rel {𝑥𝜑} → ((V × V) ∩ {𝑥𝜑}) = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
183, 5, 173eqtr3a 2679 1 (Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  {crab 2911  Vcvv 3189  cin 3558  𝒫 cpw 4135   cint 4445   × cxp 5077  ccnv 5078  Rel wrel 5084
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-int 4446  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087
This theorem is referenced by: (None)
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