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Theorem imasncls 23043
Description: If a relation graph is closed, then an image set of a singleton is also closed. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Hypotheses
Ref Expression
imasnopn.1 𝑋 = 𝐽
imasnopn.2 𝑌 = 𝐾
Assertion
Ref Expression
imasncls (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))

Proof of Theorem imasncls
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 imasnopn.2 . . . . . . 7 𝑌 = 𝐾
21toptopon 22266 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32biimpi 215 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
43ad2antlr 725 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5 imasnopn.1 . . . . . . . 8 𝑋 = 𝐽
65toptopon 22266 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
76biimpi 215 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
87ad2antrr 724 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9 simprr 771 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐴𝑋)
104, 8, 9cnmptc 23013 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐽))
114cnmptid 23012 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌𝑦) ∈ (𝐾 Cn 𝐾))
124, 10, 11cnmpt1t 23016 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13 simprl 769 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
145, 1txuni 22943 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
1514adantr 481 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
1613, 15sseqtrd 3984 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝑅 (𝐽 ×t 𝐾))
17 eqid 2736 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
1817cncls2i 22621 . . 3 (((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 (𝐽 ×t 𝐾)) → ((cls‘𝐾)‘((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)) ⊆ ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
1912, 16, 18syl2anc 584 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)) ⊆ ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
20 nfv 1917 . . . . 5 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋))
21 nfcv 2907 . . . . 5 𝑦(𝑅 “ {𝐴})
22 nfrab1 3426 . . . . 5 𝑦{𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23 imass1 6053 . . . . . . . . . . 11 (𝑅 ⊆ (𝑋 × 𝑌) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
2413, 23syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
25 xpimasn 6137 . . . . . . . . . . 11 (𝐴𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
2625ad2antll 727 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
2724, 26sseqtrd 3984 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
2827sseld 3943 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦𝑌))
2928pm4.71rd 563 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦𝑌𝑦 ∈ (𝑅 “ {𝐴}))))
30 elimasng 6040 . . . . . . . . . 10 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3130elvd 3452 . . . . . . . . 9 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3231ad2antll 727 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3332anbi2d 629 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑦𝑌𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
3429, 33bitrd 278 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
35 rabid 3427 . . . . . 6 (𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3634, 35bitr4di 288 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
3720, 21, 22, 36eqrd 3963 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
38 eqid 2736 . . . . 5 (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩)
3938mptpreima 6190 . . . 4 ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4037, 39eqtr4di 2794 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
4140fveq2d 6846 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
42 nfcv 2907 . . . 4 𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
43 nfrab1 3426 . . . 4 𝑦{𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
44 txtop 22920 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4544adantr 481 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
4617clsss3 22410 . . . . . . . . . . . 12 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 (𝐽 ×t 𝐾)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝐽 ×t 𝐾))
4745, 16, 46syl2anc 584 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝐽 ×t 𝐾))
4847, 15sseqtrrd 3985 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
49 imass1 6053 . . . . . . . . . 10 (((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
5048, 49syl 17 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
5150, 26sseqtrd 3984 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
5251sseld 3943 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦𝑌))
5352pm4.71rd 563 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦𝑌𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))))
54 elimasng 6040 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5554elvd 3452 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5655ad2antll 727 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5756anbi2d 629 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑦𝑌𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
5853, 57bitrd 278 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
59 rabid 3427 . . . . 5 (𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
6058, 59bitr4di 288 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ 𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}))
6120, 42, 43, 60eqrd 3963 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)})
6238mptpreima 6190 . . 3 ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
6361, 62eqtr4di 2794 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
6419, 41, 633sstr4d 3991 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  wss 3910  {csn 4586  cop 4592   cuni 4865  cmpt 5188   × cxp 5631  ccnv 5632  cima 5636  cfv 6496  (class class class)co 7357  Topctop 22242  TopOnctopon 22259  clsccl 22369   Cn ccn 22575   ×t ctx 22911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-cls 22372  df-cn 22578  df-cnp 22579  df-tx 22913
This theorem is referenced by:  utopreg  23604
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