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Theorem imasncld 21475
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 1841 . . . 4 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋))
2 nfcv 2762 . . . 4 𝑦(𝑅 “ {𝐴})
3 nfrab1 3117 . . . 4 𝑦{𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4 simprl 793 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)))
5 eqid 2620 . . . . . . . . . . . . 13 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
65cldss 20814 . . . . . . . . . . . 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 2620 . . . . . . . . . . . . 13 𝐾 = 𝐾
108, 9txuni 21376 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
1110adantr 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
127, 11sseqtr4d 3634 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝐾))
13 imass1 5488 . . . . . . . . . 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 5567 . . . . . . . . . 10 (𝐴𝑋 → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1615ad2antll 764 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1714, 16sseqtrd 3633 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝐾)
1817sseld 3594 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 𝐾))
1918pm4.71rd 666 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴}))))
20 vex 3198 . . . . . . . . 9 𝑦 ∈ V
21 elimasng 5479 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2220, 21mpan2 706 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2322ad2antll 764 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2423anbi2d 739 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
2519, 24bitrd 268 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
26 rabid 3111 . . . . 5 (𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2725, 26syl6bbr 278 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
281, 2, 3, 27eqrd 3614 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
29 eqid 2620 . . . 4 (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩)
3029mptpreima 5616 . . 3 ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
3128, 30syl6eqr 2672 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
329toptopon 20703 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3332biimpi 206 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
3433ad2antlr 762 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘ 𝐾))
358toptopon 20703 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3635biimpi 206 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
3736ad2antrr 761 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
38 simprr 795 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐴𝑋)
3934, 37, 38cnmptc 21446 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝐴) ∈ (𝐾 Cn 𝐽))
4034cnmptid 21445 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝑦) ∈ (𝐾 Cn 𝐾))
4134, 39, 40cnmpt1t 21449 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
42 cnclima 21053 . . 3 (((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾))) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
4341, 4, 42syl2anc 692 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
4431, 43eqeltrd 2699 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 196  wa 384   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  wss 3567  {csn 4168  cop 4174   cuni 4427  cmpt 4720   × cxp 5102  ccnv 5103  cima 5107  cfv 5876  (class class class)co 6635  Topctop 20679  TopOnctopon 20696  Clsdccld 20801   Cn ccn 21009   ×t ctx 21344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cld 20804  df-cn 21012  df-cnp 21013  df-tx 21346
This theorem is referenced by: (None)
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