Theorem cnvcnvintabd 43624

Description: Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)

Hypothesis

Ref	Expression
cnvcnvintabd.x	⊢ (𝜑 → ∃𝑥𝜓)

Assertion

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

Distinct variable groups:   𝜓,𝑤   𝑥,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑥)

Proof of Theorem cnvcnvintabd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvcnv 6181	. . . . . . . . . 10 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
2	1	eleq2i 2826	. . . . . . . . 9 ⊢ (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ (𝑥 ∩ (V × V)))
3		elin 3942	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ (V × V)))
4	3	rbaib 538	. . . . . . . . 9 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ 𝑦 ∈ 𝑥))
5	2, 4	bitrid 283	. . . . . . . 8 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ 𝑥))
6	5	bicomd 223	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ◡◡𝑥))
7	6	imbi2d 340	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((𝜓 → 𝑦 ∈ 𝑥) ↔ (𝜓 → 𝑦 ∈ ◡◡𝑥)))
8	7	albidv 1920	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (∀𝑥(𝜓 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
9	8	pm5.32i 574	. . . 4 ⊢ ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
10		cnvcnvintabd.x	. . . . . . 7 ⊢ (𝜑 → ∃𝑥𝜓)
11		pm5.5 361	. . . . . . 7 ⊢ (∃𝑥𝜓 → ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
13	12	bicomd 223	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓 → 𝑦 ∈ (V × V))))
14	13	anbi1d 631	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
15	9, 14	bitrid 283	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
16		elcnvcnvintab 43606	. . 3 ⊢ (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)))
17		vex 3463	. . . . . 6 ⊢ 𝑥 ∈ V
18		cnvexg 7920	. . . . . 6 ⊢ (𝑥 ∈ V → ◡𝑥 ∈ V)
19		cnvexg 7920	. . . . . 6 ⊢ (◡𝑥 ∈ V → ◡◡𝑥 ∈ V)
20	17, 18, 19	mp2b 10	. . . . 5 ⊢ ◡◡𝑥 ∈ V
21		relcnv 6091	. . . . . 6 ⊢ Rel ◡◡𝑥
22		df-rel 5661	. . . . . 6 ⊢ (Rel ◡◡𝑥 ↔ ◡◡𝑥 ⊆ (V × V))
23	21, 22	mpbi 230	. . . . 5 ⊢ ◡◡𝑥 ⊆ (V × V)
24	20, 23	elmapintrab 43600	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
25	24	elv 3464	. . 3 ⊢ (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
26	15, 16, 25	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)}))
27	26	eqrdv 2733	1 ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2713  {crab 3415  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∩ cint 4922   × cxp 5652  ^◡ccnv 5653  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by: (None)