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Theorem imasnopn 23713
Description: If a relation graph is open, then an image set of a singleton is also open. 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
imasnopn (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)

Proof of Theorem imasnopn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1911 . . . 4 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋))
2 nfcv 2902 . . . 4 𝑦(𝑅 “ {𝐴})
3 nfrab1 3453 . . . 4 𝑦{𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4 txtop 23592 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
54adantr 480 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
6 simprl 771 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 ∈ (𝐽 ×t 𝐾))
7 eqid 2734 . . . . . . . . . . . . 13 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
87eltopss 22928 . . . . . . . . . . . 12 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → 𝑅 (𝐽 ×t 𝐾))
95, 6, 8syl2anc 584 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 (𝐽 ×t 𝐾))
10 imasnopn.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
11 eqid 2734 . . . . . . . . . . . . 13 𝐾 = 𝐾
1210, 11txuni 23615 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
1312adantr 480 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
149, 13sseqtrrd 4036 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝐾))
15 imass1 6121 . . . . . . . . . 10 (𝑅 ⊆ (𝑋 × 𝐾) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝐾) “ {𝐴}))
1614, 15syl 17 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝐾) “ {𝐴}))
17 xpimasn 6206 . . . . . . . . . 10 (𝐴𝑋 → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1817ad2antll 729 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1916, 18sseqtrd 4035 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝐾)
2019sseld 3993 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 𝐾))
2120pm4.71rd 562 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴}))))
22 elimasng 6108 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2322elvd 3483 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2423ad2antll 729 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2524anbi2d 630 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
2621, 25bitrd 279 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
27 rabid 3454 . . . . 5 (𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2826, 27bitr4di 289 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
291, 2, 3, 28eqrd 4014 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
30 eqid 2734 . . . 4 (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩)
3130mptpreima 6259 . . 3 ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
3229, 31eqtr4di 2792 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
3311toptopon 22938 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3433biimpi 216 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
3534ad2antlr 727 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘ 𝐾))
3610toptopon 22938 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3736biimpi 216 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
3837ad2antrr 726 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
39 simprr 773 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐴𝑋)
4035, 38, 39cnmptc 23685 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾𝐴) ∈ (𝐾 Cn 𝐽))
4135cnmptid 23684 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾𝑦) ∈ (𝐾 Cn 𝐾))
4235, 40, 41cnmpt1t 23688 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
43 cnima 23288 . . 3 (((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
4442, 6, 43syl2anc 584 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
4532, 44eqeltrd 2838 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  wss 3962  {csn 4630  cop 4636   cuni 4911  cmpt 5230   × cxp 5686  ccnv 5687  cima 5691  cfv 6562  (class class class)co 7430  Topctop 22914  TopOnctopon 22931   Cn ccn 23247   ×t ctx 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cn 23250  df-cnp 23251  df-tx 23585
This theorem is referenced by: (None)
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