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Theorem clrellem 38449
 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 (𝜑𝑌 ∈ V)
clrellem.rel (𝜑 → Rel 𝑋)
clrellem.sub (𝑥 = 𝑌 → (𝜓𝜒))
clrellem.sup (𝜑𝑋𝑌)
clrellem.maj (𝜑𝜒)
Assertion
Ref Expression
clrellem (𝜑 → Rel {𝑥 ∣ (𝑋𝑥𝜓)})
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.
StepHypRef Expression
1 clrellem.y . . . 4 (𝜑𝑌 ∈ V)
2 cnvexg 7278 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
3 cnvexg 7278 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
41, 2, 33syl 18 . . 3 (𝜑𝑌 ∈ V)
5 clrellem.rel . . . . . 6 (𝜑 → Rel 𝑋)
6 dfrel2 5741 . . . . . 6 (Rel 𝑋𝑋 = 𝑋)
75, 6sylib 208 . . . . 5 (𝜑𝑋 = 𝑋)
8 clrellem.sup . . . . . 6 (𝜑𝑋𝑌)
9 cnvss 5450 . . . . . 6 (𝑋𝑌𝑋𝑌)
10 cnvss 5450 . . . . . 6 (𝑋𝑌𝑋𝑌)
118, 9, 103syl 18 . . . . 5 (𝜑𝑋𝑌)
127, 11eqsstr3d 3781 . . . 4 (𝜑𝑋𝑌)
13 clrellem.maj . . . 4 (𝜑𝜒)
14 relcnv 5661 . . . . 5 Rel 𝑌
1514a1i 11 . . . 4 (𝜑 → Rel 𝑌)
1612, 13, 15jca31 558 . . 3 (𝜑 → ((𝑋𝑌𝜒) ∧ Rel 𝑌))
17 clrellem.sub . . . . 5 (𝑥 = 𝑌 → (𝜓𝜒))
1817cleq2lem 38434 . . . 4 (𝑥 = 𝑌 → ((𝑋𝑥𝜓) ↔ (𝑋𝑌𝜒)))
19 releq 5358 . . . 4 (𝑥 = 𝑌 → (Rel 𝑥 ↔ Rel 𝑌))
2018, 19anbi12d 749 . . 3 (𝑥 = 𝑌 → (((𝑋𝑥𝜓) ∧ Rel 𝑥) ↔ ((𝑋𝑌𝜒) ∧ Rel 𝑌)))
214, 16, 20elabd 3492 . 2 (𝜑 → ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
22 releq 5358 . . . 4 (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
2322rexab2 3514 . . 3 (∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
2423biimpri 218 . 2 (∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦)
25 relint 5398 . 2 (∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦 → Rel {𝑥 ∣ (𝑋𝑥𝜓)})
2621, 24, 253syl 18 1 (𝜑 → Rel {𝑥 ∣ (𝑋𝑥𝜓)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632  ∃wex 1853   ∈ wcel 2139  {cab 2746  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  ∩ cint 4627  ◡ccnv 5265  Rel wrel 5271 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iin 4675  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277 This theorem is referenced by: (None)
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