Theorem relintab 42334

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 6192	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4202	. . 3 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2761	. 2 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4		dfrel2 6189	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} ↔ ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
5	4	biimpi 215	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
6		relintabex 42332	. . . 4 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)
7	6	xpinintabd 42331	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8		incom 4202	. . . . . . . . 9 ⊢ ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9		cnvcnv 6192	. . . . . . . . 9 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
10	8, 9	eqtr4i 2764	. . . . . . . 8 ⊢ ((V × V) ∩ 𝑥) = ◡◡𝑥
11	10	eqeq2i 2746	. . . . . . 7 ⊢ (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = ◡◡𝑥)
12	11	anbi1i 625	. . . . . 6 ⊢ ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = ◡◡𝑥 ∧ 𝜑))
13	12	exbii 1851	. . . . 5 ⊢ (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑))
14	13	rabbii 3439	. . . 4 ⊢ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
15	14	inteqi 4955	. . 3 ⊢ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
16	7, 15	eqtrdi 2789	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})
17	3, 5, 16	3eqtr3a 2797	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782  {cab 2710  {crab 3433  Vcvv 3475   ∩ cin 3948  𝒫 cpw 4603  ∩ cint 4951   × cxp 5675  ^◡ccnv 5676  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-int 4952  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685

This theorem is referenced by: (None)