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Theorem imasncld 21242
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 1828 . . . 4 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋))
2 nfcv 2746 . . . 4 𝑦(𝑅 “ {𝐴})
3 nfrab1 3094 . . . 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 2605 . . . . . . . . . . . . 13 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
65cldss 20581 . . . . . . . . . . . 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 2605 . . . . . . . . . . . . 13 𝐾 = 𝐾
108, 9txuni 21143 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
1110adantr 479 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
127, 11sseqtr4d 3600 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝐾))
13 imass1 5402 . . . . . . . . . 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 5480 . . . . . . . . . 10 (𝐴𝑋 → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1615ad2antll 760 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1714, 16sseqtrd 3599 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝐾)
1817sseld 3562 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 𝐾))
1918pm4.71rd 664 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴}))))
20 vex 3171 . . . . . . . . 9 𝑦 ∈ V
21 elimasng 5393 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2220, 21mpan2 702 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2322ad2antll 760 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2423anbi2d 735 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
2519, 24bitrd 266 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
26 rabid 3090 . . . . 5 (𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2725, 26syl6bbr 276 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
281, 2, 3, 27eqrd 3581 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
29 eqid 2605 . . . 4 (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩)
3029mptpreima 5527 . . 3 ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
3128, 30syl6eqr 2657 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
329toptopon 20486 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3332biimpi 204 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
3433ad2antlr 758 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘ 𝐾))
358toptopon 20486 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3635biimpi 204 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
3736ad2antrr 757 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
38 simprr 791 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → 𝐴𝑋)
3934, 37, 38cnmptc 21213 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝐴) ∈ (𝐾 Cn 𝐽))
4034cnmptid 21212 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾𝑦) ∈ (𝐾 Cn 𝐾))
4134, 39, 40cnmpt1t 21216 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
42 cnclima 20820 . . 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 690 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴𝑋)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
4431, 43eqeltrd 2683 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 194  wa 382   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535  {csn 4120  cop 4126   cuni 4362  cmpt 4633   × cxp 5022  ccnv 5023  cima 5027  cfv 5786  (class class class)co 6523  Topctop 20455  TopOnctopon 20456  Clsdccld 20568   Cn ccn 20776   ×t ctx 21111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-map 7719  df-topgen 15869  df-top 20459  df-bases 20460  df-topon 20461  df-cld 20571  df-cn 20779  df-cnp 20780  df-tx 21113
This theorem is referenced by: (None)
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