Theorem cnvcnvintabd 40300

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 6016	. . . . . . . . . 10 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
2	1	eleq2i 2881	. . . . . . . . 9 ⊢ (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ (𝑥 ∩ (V × V)))
3		elin 3897	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ (V × V)))
4	3	rbaib 542	. . . . . . . . 9 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ 𝑦 ∈ 𝑥))
5	2, 4	syl5bb 286	. . . . . . . 8 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ 𝑥))
6	5	bicomd 226	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ◡◡𝑥))
7	6	imbi2d 344	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((𝜓 → 𝑦 ∈ 𝑥) ↔ (𝜓 → 𝑦 ∈ ◡◡𝑥)))
8	7	albidv 1921	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (∀𝑥(𝜓 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
9	8	pm5.32i 578	. . . 4 ⊢ ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
10		cnvcnvintabd.x	. . . . . . 7 ⊢ (𝜑 → ∃𝑥𝜓)
11		pm5.5 365	. . . . . . 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 226	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓 → 𝑦 ∈ (V × V))))
14	13	anbi1d 632	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
15	9, 14	syl5bb 286	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
16		elcnvcnvintab 40282	. . 3 ⊢ (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)))
17		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
18		cnvexg 7611	. . . . . 6 ⊢ (𝑥 ∈ V → ◡𝑥 ∈ V)
19		cnvexg 7611	. . . . . 6 ⊢ (◡𝑥 ∈ V → ◡◡𝑥 ∈ V)
20	17, 18, 19	mp2b 10	. . . . 5 ⊢ ◡◡𝑥 ∈ V
21		relcnv 5934	. . . . . 6 ⊢ Rel ◡◡𝑥
22		df-rel 5526	. . . . . 6 ⊢ (Rel ◡◡𝑥 ↔ ◡◡𝑥 ⊆ (V × V))
23	21, 22	mpbi 233	. . . . 5 ⊢ ◡◡𝑥 ⊆ (V × V)
24	20, 23	elmapintrab 40276	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
25	24	elv 3446	. . 3 ⊢ (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
26	15, 16, 25	3bitr4g 317	. 2 ⊢ (𝜑 → (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)}))
27	26	eqrdv 2796	1 ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∩ cint 4838   × cxp 5517  ^◡ccnv 5518  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by: (None)