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Theorem clrellem 41230
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 7771 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
3 cnvexg 7771 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
41, 2, 33syl 18 . . 3 (𝜑𝑌 ∈ V)
5 clrellem.rel . . . . . 6 (𝜑 → Rel 𝑋)
6 dfrel2 6092 . . . . . 6 (Rel 𝑋𝑋 = 𝑋)
75, 6sylib 217 . . . . 5 (𝜑𝑋 = 𝑋)
8 clrellem.sup . . . . . 6 (𝜑𝑋𝑌)
9 cnvss 5781 . . . . . 6 (𝑋𝑌𝑋𝑌)
10 cnvss 5781 . . . . . 6 (𝑋𝑌𝑋𝑌)
118, 9, 103syl 18 . . . . 5 (𝜑𝑋𝑌)
127, 11eqsstrrd 3960 . . . 4 (𝜑𝑋𝑌)
13 clrellem.maj . . . 4 (𝜑𝜒)
14 relcnv 6012 . . . . 5 Rel 𝑌
1514a1i 11 . . . 4 (𝜑 → Rel 𝑌)
1612, 13, 15jca31 515 . . 3 (𝜑 → ((𝑋𝑌𝜒) ∧ Rel 𝑌))
17 clrellem.sub . . . . 5 (𝑥 = 𝑌 → (𝜓𝜒))
1817cleq2lem 41216 . . . 4 (𝑥 = 𝑌 → ((𝑋𝑥𝜓) ↔ (𝑋𝑌𝜒)))
19 releq 5687 . . . 4 (𝑥 = 𝑌 → (Rel 𝑥 ↔ Rel 𝑌))
2018, 19anbi12d 631 . . 3 (𝑥 = 𝑌 → (((𝑋𝑥𝜓) ∧ Rel 𝑥) ↔ ((𝑋𝑌𝜒) ∧ Rel 𝑌)))
214, 16, 20spcedv 3537 . 2 (𝜑 → ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
22 releq 5687 . . . 4 (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
2322rexab2 3636 . . 3 (∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
2423biimpri 227 . 2 (∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦)
25 relint 5729 . 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 205  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065  Vcvv 3432  wss 3887   cint 4879  ccnv 5588  Rel wrel 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iin 4927  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600
This theorem is referenced by: (None)
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