Theorem clrellem 40759

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 7648	. . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V)
3		cnvexg 7648	. . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V)
5		clrellem.rel	. . . . . 6 ⊢ (𝜑 → Rel 𝑋)
6		dfrel2 6015	. . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋)
7	5, 6	sylib 221	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋)
8		clrellem.sup	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
9		cnvss 5709	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌)
10		cnvss 5709	. . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌)
12	7, 11	eqsstrrd 3914	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌)
13		clrellem.maj	. . . 4 ⊢ (𝜑 → 𝜒)
14		relcnv 5935	. . . . 5 ⊢ Rel ◡◡𝑌
15	14	a1i 11	. . . 4 ⊢ (𝜑 → Rel ◡◡𝑌)
16	12, 13, 15	jca31 518	. . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))
17		clrellem.sub	. . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))
18	17	cleq2lem 40745	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒)))
19		releq 5616	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌))
20	18, 19	anbi12d 634	. . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)))
21	4, 16, 20	spcedv 3500	. 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
22		releq 5616	. . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
23	22	rexab2 3597	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
24	23	biimpri 231	. 2 ⊢ (∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦)
25		relint 5657	. 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 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2113  {cab 2716  ∃wrex 3054  Vcvv 3397   ⊆ wss 3841  ∩ cint 4833  ^◡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 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-int 4834  df-iin 4881  df-br 5028  df-opab 5090  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by: (None)