Theorem relintab 38723

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 5827	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4032	. . 3 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2849	. 2 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4		dfrel2 5824	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} ↔ ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
5	4	biimpi 208	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
6		relintabex 38721	. . . 4 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)
7	6	xpinintabd 38720	. . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)})
8		incom 4032	. . . . . . . . 9 ⊢ ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V))
9		cnvcnv 5827	. . . . . . . . 9 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
10	8, 9	eqtr4i 2852	. . . . . . . 8 ⊢ ((V × V) ∩ 𝑥) = ◡◡𝑥
11	10	eqeq2i 2837	. . . . . . 7 ⊢ (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = ◡◡𝑥)
12	11	anbi1i 617	. . . . . 6 ⊢ ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = ◡◡𝑥 ∧ 𝜑))
13	12	exbii 1947	. . . . 5 ⊢ (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑))
14	13	rabbii 3398	. . . 4 ⊢ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
15	14	inteqi 4701	. . 3 ⊢ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}
16	7, 15	syl6eq 2877	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})
17	3, 5, 16	3eqtr3a 2885	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656  ∃wex 1878  {cab 2811  {crab 3121  Vcvv 3414   ∩ cin 3797  𝒫 cpw 4378  ∩ cint 4697   × cxp 5340  ^◡ccnv 5341  Rel wrel 5347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-int 4698  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350

This theorem is referenced by: (None)