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Theorem imasncld 21865
 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.)
Hypothesis
Ref Expression
imasnopn.1 𝑋 = 𝐽
Assertion
Ref Expression
imasncld (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))

Proof of Theorem imasncld
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 2015 . . . 4 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋))
2 nfcv 2969 . . . 4 𝑦(𝑅 “ {𝐴})
3 nfrab1 3333 . . . 4 𝑦{𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4 simprl 789 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)))
5 eqid 2825 . . . . . . . . . . . . 13 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
65cldss 21204 . . . . . . . . . . . 12 (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) → 𝑅 (𝐽 ×t 𝐾))
74, 6syl 17 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 (𝐽 ×t 𝐾))
8 imasnopn.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
9 eqid 2825 . . . . . . . . . . . . 13 𝐾 = 𝐾
108, 9txuni 21766 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
1110adantr 474 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
127, 11sseqtr4d 3867 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝐾))
13 imass1 5741 . . . . . . . . . 10 (𝑅 ⊆ (𝑋 × 𝐾) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝐾) “ {𝐴}))
1412, 13syl 17 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝐾) “ {𝐴}))
15 xpimasn 5820 . . . . . . . . . 10 (𝐴𝑋 → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1615ad2antll 722 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1714, 16sseqtrd 3866 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝐾)
1817sseld 3826 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 𝐾))
1918pm4.71rd 560 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴}))))
20 elimasng 5732 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2120elvd 3419 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2221ad2antll 722 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2322anbi2d 624 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
2419, 23bitrd 271 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
25 rabid 3326 . . . . 5 (𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2624, 25syl6bbr 281 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
271, 2, 3, 26eqrd 3846 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
28 eqid 2825 . . . 4 (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩)
2928mptpreima 5869 . . 3 ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
3027, 29syl6eqr 2879 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
319toptopon 21092 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3231biimpi 208 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
3332ad2antlr 720 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘ 𝐾))
348toptopon 21092 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3534biimpi 208 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
3635ad2antrr 719 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
37 simprr 791 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐴𝑋)
3833, 36, 37cnmptc 21836 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝐴) ∈ (𝐾 Cn 𝐽))
3933cnmptid 21835 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝑦) ∈ (𝐾 Cn 𝐾))
4033, 38, 39cnmpt1t 21839 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
41 cnclima 21443 . . 3 (((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾))) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
4240, 4, 41syl2anc 581 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
4330, 42eqeltrd 2906 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3121  Vcvv 3414   ⊆ wss 3798  {csn 4397  ⟨cop 4403  ∪ cuni 4658   ↦ cmpt 4952   × cxp 5340  ◡ccnv 5341   “ cima 5345  ‘cfv 6123  (class class class)co 6905  Topctop 21068  TopOnctopon 21085  Clsdccld 21191   Cn ccn 21399   ×t ctx 21734 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-topgen 16457  df-top 21069  df-topon 21086  df-bases 21121  df-cld 21194  df-cn 21402  df-cnp 21403  df-tx 21736 This theorem is referenced by: (None)
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