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Theorem clrellem 39850
 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 7620 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
3 cnvexg 7620 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
41, 2, 33syl 18 . . 3 (𝜑𝑌 ∈ V)
5 clrellem.rel . . . . . 6 (𝜑 → Rel 𝑋)
6 dfrel2 6043 . . . . . 6 (Rel 𝑋𝑋 = 𝑋)
75, 6sylib 219 . . . . 5 (𝜑𝑋 = 𝑋)
8 clrellem.sup . . . . . 6 (𝜑𝑋𝑌)
9 cnvss 5741 . . . . . 6 (𝑋𝑌𝑋𝑌)
10 cnvss 5741 . . . . . 6 (𝑋𝑌𝑋𝑌)
118, 9, 103syl 18 . . . . 5 (𝜑𝑋𝑌)
127, 11eqsstrrd 4009 . . . 4 (𝜑𝑋𝑌)
13 clrellem.maj . . . 4 (𝜑𝜒)
14 relcnv 5964 . . . . 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 39836 . . . 4 (𝑥 = 𝑌 → ((𝑋𝑥𝜓) ↔ (𝑋𝑌𝜒)))
19 releq 5649 . . . 4 (𝑥 = 𝑌 → (Rel 𝑥 ↔ Rel 𝑌))
2018, 19anbi12d 630 . . 3 (𝑥 = 𝑌 → (((𝑋𝑥𝜓) ∧ Rel 𝑥) ↔ ((𝑋𝑌𝜒) ∧ Rel 𝑌)))
214, 16, 20elabd 3671 . 2 (𝜑 → ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
22 releq 5649 . . . 4 (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
2322rexab2 3694 . . 3 (∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
2423biimpri 229 . 2 (∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦)
25 relint 5690 . 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 207   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2803  ∃wrex 3143  Vcvv 3499   ⊆ wss 3939  ∩ cint 4873  ◡ccnv 5552  Rel wrel 5558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-iin 4919  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564 This theorem is referenced by: (None)
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