Theorem cnvcnvintabd 44059

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 6147	. . . . . . . . . 10 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V))
2	1	eleq2i 2833	. . . . . . . . 9 ⊢ (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ (𝑥 ∩ (V × V)))
3		elin 3901	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ (V × V)))
4	3	rbaib 544	. . . . . . . . 9 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ (𝑥 ∩ (V × V)) ↔ 𝑦 ∈ 𝑥))
5	2, 4	bitrid 285	. . . . . . . 8 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ ◡◡𝑥 ↔ 𝑦 ∈ 𝑥))
6	5	bicomd 225	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ◡◡𝑥))
7	6	imbi2d 342	. . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((𝜓 → 𝑦 ∈ 𝑥) ↔ (𝜓 → 𝑦 ∈ ◡◡𝑥)))
8	7	albidv 1928	. . . . 5 ⊢ (𝑦 ∈ (V × V) → (∀𝑥(𝜓 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
9	8	pm5.32i 580	. . . 4 ⊢ ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)))
10		cnvcnvintabd.x	. . . . . . 7 ⊢ (𝜑 → ∃𝑥𝜓)
11		pm5.5 363	. . . . . . 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 638	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
15	9, 14	bitrid 285	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
16		elcnvcnvintab 44041	. . 3 ⊢ (𝑦 ∈ ◡◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ 𝑥)))
17		vex 3437	. . . . . 6 ⊢ 𝑥 ∈ V
18		cnvexg 7868	. . . . . 6 ⊢ (𝑥 ∈ V → ◡𝑥 ∈ V)
19		cnvexg 7868	. . . . . 6 ⊢ (◡𝑥 ∈ V → ◡◡𝑥 ∈ V)
20	17, 18, 19	mp2b 10	. . . . 5 ⊢ ◡◡𝑥 ∈ V
21		relcnv 6063	. . . . . 6 ⊢ Rel ◡◡𝑥
22		df-rel 5628	. . . . . 6 ⊢ (Rel ◡◡𝑥 ↔ ◡◡𝑥 ⊆ (V × V))
23	21, 22	mpbi 232	. . . . 5 ⊢ ◡◡𝑥 ⊆ (V × V)
24	20, 23	elmapintrab 44035	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡◡𝑥))))
25	24	elv 3438	. . 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 2739	1 ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719  {crab 3393  Vcvv 3433   ∩ cin 3884   ⊆ wss 3885  𝒫 cpw 4532  ∩ cint 4880   × cxp 5619  ^◡ccnv 5620  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by: (None)