Theorem clrellem 43899

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 7868	. . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V)
3		cnvexg 7868	. . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V)
5		clrellem.rel	. . . . . 6 ⊢ (𝜑 → Rel 𝑋)
6		dfrel2 6148	. . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋)
7	5, 6	sylib 218	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋)
8		clrellem.sup	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
9		cnvss 5822	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌)
10		cnvss 5822	. . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌)
12	7, 11	eqsstrrd 3970	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌)
13		clrellem.maj	. . . 4 ⊢ (𝜑 → 𝜒)
14		relcnv 6064	. . . . 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 43885	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒)))
19		releq 5727	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌))
20	18, 19	anbi12d 633	. . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)))
21	4, 16, 20	spcedv 3553	. 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
22		releq 5727	. . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
23	22	rexab2 3658	. . 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 5769	. 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∃wrex 3061  Vcvv 3441   ⊆ wss 3902  ∩ cint 4903  ^◡ccnv 5624  Rel wrel 5630

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iin 4950  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by: (None)