Theorem cnvcnvintabd 39953

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 6043	. . . . . . . . . 10 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
2	1	eleq2i 2904	. . . . . . . . 9 ⊢ (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ (𝑥 ∩ (V × V)))
3		elin 4168	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ (V × V)))
4	3	rbaib 541	. . . . . . . . 9 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ 𝑦 ∈ 𝑥))
5	2, 4	syl5bb 285	. . . . . . . 8 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ 𝑥))
6	5	bicomd 225	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ◡◡𝑥))
7	6	imbi2d 343	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((𝜓 → 𝑦 ∈ 𝑥) ↔ (𝜓 → 𝑦 ∈ ◡◡𝑥)))
8	7	albidv 1917	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (∀𝑥(𝜓 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
9	8	pm5.32i 577	. . . 4 ⊢ ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
10		cnvcnvintabd.x	. . . . . . 7 ⊢ (𝜑 → ∃𝑥𝜓)
11		pm5.5 364	. . . . . . 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 225	. . . . 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	syl5bb 285	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
16		elcnvcnvintab 39935	. . 3 ⊢ (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)))
17		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
18		cnvexg 7623	. . . . . 6 ⊢ (𝑥 ∈ V → ◡𝑥 ∈ V)
19		cnvexg 7623	. . . . . 6 ⊢ (◡𝑥 ∈ V → ◡◡𝑥 ∈ V)
20	17, 18, 19	mp2b 10	. . . . 5 ⊢ ◡◡𝑥 ∈ V
21		relcnv 5961	. . . . . 6 ⊢ Rel ◡◡𝑥
22		df-rel 5556	. . . . . 6 ⊢ (Rel ◡◡𝑥 ↔ ◡◡𝑥 ⊆ (V × V))
23	21, 22	mpbi 232	. . . . 5 ⊢ ◡◡𝑥 ⊆ (V × V)
24	20, 23	elmapintrab 39929	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
25	24	elv 3499	. . 3 ⊢ (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
26	15, 16, 25	3bitr4g 316	. 2 ⊢ (𝜑 → (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)}))
27	26	eqrdv 2819	1 ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  {crab 3142  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ∩ cint 4868   × cxp 5547  ^◡ccnv 5548  Rel wrel 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560

This theorem is referenced by: (None)