Theorem clrellem 44162

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 7901	. . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V)
3		cnvexg 7901	. . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V)
5		clrellem.rel	. . . . . 6 ⊢ (𝜑 → Rel 𝑋)
6		dfrel2 6171	. . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋)
7	5, 6	sylib 220	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋)
8		clrellem.sup	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
9		cnvss 5842	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌)
10		cnvss 5842	. . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌)
12	7, 11	eqsstrrd 3971	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌)
13		clrellem.maj	. . . 4 ⊢ (𝜑 → 𝜒)
14		relcnv 6090	. . . . 5 ⊢ Rel ◡◡𝑌
15	14	a1i 11	. . . 4 ⊢ (𝜑 → Rel ◡◡𝑌)
16	12, 13, 15	jca31 522	. . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))
17		clrellem.sub	. . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))
18	17	cleq2lem 44148	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒)))
19		releq 5747	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌))
20	18, 19	anbi12d 641	. . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)))
21	4, 16, 20	spcedv 3557	. 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
22		releq 5747	. . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
23	22	rexab2 3661	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
24	23	biimpri 230	. 2 ⊢ (∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦)
25		relint 5790	. 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 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∃wrex 3085  Vcvv 3453   ⊆ wss 3904  ∩ cint 4904  ^◡ccnv 5644  Rel wrel 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iin 4951  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656

This theorem is referenced by: (None)