Theorem relintab 44034

Description: Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)

Assertion

Ref	Expression
relintab	⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})

Distinct variable groups:   𝜑,𝑤   𝑥,𝑤

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem relintab

Step	Hyp	Ref	Expression
1		cnvcnv 6150	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4145	. . 3 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2763	. 2 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4		dfrel2 6147	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} ↔ ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
5	4	biimpi 217	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
6		relintabex 44032	. . . 4 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)
7	6	xpinintabd 44031	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8		incom 4145	. . . . . . . . 9 ⊢ ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9		cnvcnv 6150	. . . . . . . . 9 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
10	8, 9	eqtr4i 2766	. . . . . . . 8 ⊢ ((V × V) ∩ 𝑥) = ◡◡𝑥
11	10	eqeq2i 2753	. . . . . . 7 ⊢ (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = ◡◡𝑥)
12	11	anbi1i 630	. . . . . 6 ⊢ ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = ◡◡𝑥 ∧ 𝜑))
13	12	exbii 1855	. . . . 5 ⊢ (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑))
14	13	rabbii 3397	. . . 4 ⊢ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
15	14	inteqi 4888	. . 3 ⊢ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
16	7, 15	eqtrdi 2791	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})
17	3, 5, 16	3eqtr3a 2799	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786  {cab 2718  {crab 3392  Vcvv 3432   ∩ cin 3889  𝒫 cpw 4536  ∩ cint 4884   × cxp 5623  ^◡ccnv 5624  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-int 4885  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633

This theorem is referenced by: (None)