Theorem relintab 43700

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 6144	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4158	. . 3 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2756	. 2 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4		dfrel2 6141	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} ↔ ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
5	4	biimpi 216	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
6		relintabex 43698	. . . 4 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)
7	6	xpinintabd 43697	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8		incom 4158	. . . . . . . . 9 ⊢ ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9		cnvcnv 6144	. . . . . . . . 9 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
10	8, 9	eqtr4i 2759	. . . . . . . 8 ⊢ ((V × V) ∩ 𝑥) = ◡◡𝑥
11	10	eqeq2i 2746	. . . . . . 7 ⊢ (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = ◡◡𝑥)
12	11	anbi1i 624	. . . . . 6 ⊢ ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = ◡◡𝑥 ∧ 𝜑))
13	12	exbii 1849	. . . . 5 ⊢ (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑))
14	13	rabbii 3401	. . . 4 ⊢ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
15	14	inteqi 4901	. . 3 ⊢ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
16	7, 15	eqtrdi 2784	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})
17	3, 5, 16	3eqtr3a 2792	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780  {cab 2711  {crab 3396  Vcvv 3437   ∩ cin 3897  𝒫 cpw 4549  ∩ cint 4897   × cxp 5617  ^◡ccnv 5618  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-int 4898  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627

This theorem is referenced by: (None)