Theorem clrellem 43635

Description: When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)

Hypotheses

Ref	Expression
clrellem.y	⊢ (𝜑 → 𝑌 ∈ V)
clrellem.rel	⊢ (𝜑 → Rel 𝑋)
clrellem.sub	⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))
clrellem.sup	⊢ (𝜑 → 𝑋 ⊆ 𝑌)
clrellem.maj	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
clrellem	⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})

Distinct variable groups:   𝑥,𝑋   𝑥,𝑌   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem clrellem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clrellem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
2		cnvexg 7946	. . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V)
3		cnvexg 7946	. . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V)
5		clrellem.rel	. . . . . 6 ⊢ (𝜑 → Rel 𝑋)
6		dfrel2 6209	. . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋)
7	5, 6	sylib 218	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋)
8		clrellem.sup	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
9		cnvss 5883	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌)
10		cnvss 5883	. . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌)
12	7, 11	eqsstrrd 4019	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌)
13		clrellem.maj	. . . 4 ⊢ (𝜑 → 𝜒)
14		relcnv 6122	. . . . 5 ⊢ Rel ◡◡𝑌
15	14	a1i 11	. . . 4 ⊢ (𝜑 → Rel ◡◡𝑌)
16	12, 13, 15	jca31 514	. . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))
17		clrellem.sub	. . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))
18	17	cleq2lem 43621	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒)))
19		releq 5786	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌))
20	18, 19	anbi12d 632	. . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)))
21	4, 16, 20	spcedv 3598	. 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
22		releq 5786	. . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
23	22	rexab2 3705	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
24	23	biimpri 228	. 2 ⊢ (∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦)
25		relint 5829	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
26	21, 24, 25	3syl 18	1 ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∩ cint 4946  ^◡ccnv 5684  Rel wrel 5690

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by: (None)