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Theorem clrellem 40759
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 7648 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
3 cnvexg 7648 . . . 4 (𝑌 ∈ V → 𝑌 ∈ V)
41, 2, 33syl 18 . . 3 (𝜑𝑌 ∈ V)
5 clrellem.rel . . . . . 6 (𝜑 → Rel 𝑋)
6 dfrel2 6015 . . . . . 6 (Rel 𝑋𝑋 = 𝑋)
75, 6sylib 221 . . . . 5 (𝜑𝑋 = 𝑋)
8 clrellem.sup . . . . . 6 (𝜑𝑋𝑌)
9 cnvss 5709 . . . . . 6 (𝑋𝑌𝑋𝑌)
10 cnvss 5709 . . . . . 6 (𝑋𝑌𝑋𝑌)
118, 9, 103syl 18 . . . . 5 (𝜑𝑋𝑌)
127, 11eqsstrrd 3914 . . . 4 (𝜑𝑋𝑌)
13 clrellem.maj . . . 4 (𝜑𝜒)
14 relcnv 5935 . . . . 5 Rel 𝑌
1514a1i 11 . . . 4 (𝜑 → Rel 𝑌)
1612, 13, 15jca31 518 . . 3 (𝜑 → ((𝑋𝑌𝜒) ∧ Rel 𝑌))
17 clrellem.sub . . . . 5 (𝑥 = 𝑌 → (𝜓𝜒))
1817cleq2lem 40745 . . . 4 (𝑥 = 𝑌 → ((𝑋𝑥𝜓) ↔ (𝑋𝑌𝜒)))
19 releq 5616 . . . 4 (𝑥 = 𝑌 → (Rel 𝑥 ↔ Rel 𝑌))
2018, 19anbi12d 634 . . 3 (𝑥 = 𝑌 → (((𝑋𝑥𝜓) ∧ Rel 𝑥) ↔ ((𝑋𝑌𝜒) ∧ Rel 𝑌)))
214, 16, 20spcedv 3500 . 2 (𝜑 → ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
22 releq 5616 . . . 4 (𝑦 = 𝑥 → (Rel 𝑦 ↔ Rel 𝑥))
2322rexab2 3597 . . 3 (∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦 ↔ ∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥))
2423biimpri 231 . 2 (∃𝑥((𝑋𝑥𝜓) ∧ Rel 𝑥) → ∃𝑦 ∈ {𝑥 ∣ (𝑋𝑥𝜓)}Rel 𝑦)
25 relint 5657 . 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 209  wa 399   = wceq 1542  wex 1786  wcel 2113  {cab 2716  wrex 3054  Vcvv 3397  wss 3841   cint 4833  ccnv 5518  Rel wrel 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-int 4834  df-iin 4881  df-br 5028  df-opab 5090  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530
This theorem is referenced by: (None)
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