Theorem clrellem 44210

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 7909	. . . 4 ⊢ (𝑌 ∈ V → ◡𝑌 ∈ V)
3		cnvexg 7909	. . . 4 ⊢ (◡𝑌 ∈ V → ◡◡𝑌 ∈ V)
4	1, 2, 3	3syl 19	. . 3 ⊢ (𝜑 → ◡◡𝑌 ∈ V)
5		clrellem.rel	. . . . . 6 ⊢ (𝜑 → Rel 𝑋)
6		dfrel2 6179	. . . . . 6 ⊢ (Rel 𝑋 ↔ ◡◡𝑋 = 𝑋)
7	5, 6	sylib 221	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 = 𝑋)
8		clrellem.sup	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
9		cnvss 5849	. . . . . 6 ⊢ (𝑋 ⊆ 𝑌 → ◡𝑋 ⊆ ◡𝑌)
10		cnvss 5849	. . . . . 6 ⊢ (◡𝑋 ⊆ ◡𝑌 → ◡◡𝑋 ⊆ ◡◡𝑌)
11	8, 9, 10	3syl 19	. . . . 5 ⊢ (𝜑 → ◡◡𝑋 ⊆ ◡◡𝑌)
12	7, 11	eqsstrrd 3974	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ◡◡𝑌)
13		clrellem.maj	. . . 4 ⊢ (𝜑 → 𝜒)
14		relcnv 6097	. . . . 5 ⊢ Rel ◡◡𝑌
15	14	a1i 11	. . . 4 ⊢ (𝜑 → Rel ◡◡𝑌)
16	12, 13, 15	jca31 523	. . 3 ⊢ (𝜑 → ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌))
17		clrellem.sub	. . . . 5 ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))
18	17	cleq2lem 44196	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ ◡◡𝑌 ∧ 𝜒)))
19		releq 5754	. . . 4 ⊢ (𝑥 = ◡◡𝑌 → (Rel 𝑥 ↔ Rel ◡◡𝑌))
20	18, 19	anbi12d 643	. . 3 ⊢ (𝑥 = ◡◡𝑌 → (((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥) ↔ ((𝑋 ⊆ ◡◡𝑌 ∧ 𝜒) ∧ Rel ◡◡𝑌)))
21	4, 16, 20	spcedv 3560	. 2 ⊢ (𝜑 → ∃𝑥((𝑋 ⊆ 𝑥 ∧ 𝜓) ∧ Rel 𝑥))
22		releq 5754	. . . 4 ⊢ (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
23	22	rexab2 3665	. . 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 5797	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)}Rel 𝑦 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
26	21, 24, 25	3syl 19	1 ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ∩ cint 4908  ^◡ccnv 5651  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iin 4955  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by: (None)