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Theorem imasncls 21543
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 20770 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32biimpi 206 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
43ad2antlr 763 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5 imasnopn.1 . . . . . . . 8 𝑋 = 𝐽
65toptopon 20770 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
76biimpi 206 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
87ad2antrr 762 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9 simprr 811 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝐴𝑋)
104, 8, 9cnmptc 21513 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐽))
114cnmptid 21512 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌𝑦) ∈ (𝐾 Cn 𝐾))
124, 10, 11cnmpt1t 21516 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13 simprl 809 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
145, 1txuni 21443 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
1514adantr 480 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
1613, 15sseqtrd 3674 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → 𝑅 (𝐽 ×t 𝐾))
17 eqid 2651 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
1817cncls2i 21122 . . 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 694 . 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 1883 . . . . 5 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋))
21 nfcv 2793 . . . . 5 𝑦(𝑅 “ {𝐴})
22 nfrab1 3152 . . . . 5 𝑦{𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23 imass1 5535 . . . . . . . . . . 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 5614 . . . . . . . . . . 11 (𝐴𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
2625ad2antll 765 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
2724, 26sseqtrd 3674 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
2827sseld 3635 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦𝑌))
2928pm4.71rd 668 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦𝑌𝑦 ∈ (𝑅 “ {𝐴}))))
30 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
31 elimasng 5526 . . . . . . . . . 10 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3230, 31mpan2 707 . . . . . . . . 9 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3332ad2antll 765 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3433anbi2d 740 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑦𝑌𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
3529, 34bitrd 268 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
36 rabid 3145 . . . . . 6 (𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
3735, 36syl6bbr 278 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
3820, 21, 22, 37eqrd 3655 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
39 eqid 2651 . . . . 5 (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩)
4039mptpreima 5666 . . . 4 ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4138, 40syl6eqr 2703 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
4241fveq2d 6233 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
43 nfcv 2793 . . . 4 𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
44 nfrab1 3152 . . . 4 𝑦{𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
45 txtop 21420 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
4645adantr 480 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
4717clsss3 20911 . . . . . . . . . . . 12 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 (𝐽 ×t 𝐾)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝐽 ×t 𝐾))
4846, 16, 47syl2anc 694 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝐽 ×t 𝐾))
4948, 15sseqtr4d 3675 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
50 imass1 5535 . . . . . . . . . 10 (((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
5149, 50syl 17 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
5251, 26sseqtrd 3674 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
5352sseld 3635 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦𝑌))
5453pm4.71rd 668 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦𝑌𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))))
55 elimasng 5526 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5630, 55mpan2 707 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5756ad2antll 765 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
5857anbi2d 740 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → ((𝑦𝑌𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
5954, 58bitrd 268 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
60 rabid 3145 . . . . 5 (𝑦 ∈ {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
6159, 60syl6bbr 278 . . . 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)}))
6220, 43, 44, 61eqrd 3655 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)})
6339mptpreima 5666 . . 3 ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
6462, 63syl6eqr 2703 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = ((𝑦𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
6519, 42, 643sstr4d 3681 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 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  {csn 4210  cop 4216   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  cima 5146  cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763  clsccl 20870   Cn ccn 21076   ×t ctx 21411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-cls 20873  df-cn 21079  df-cnp 21080  df-tx 21413
This theorem is referenced by:  utopreg  22103
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