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Theorem imasnopn 21245
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 1829 . . . 4 𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋))
2 nfcv 2750 . . . 4 𝑦(𝑅 “ {𝐴})
3 nfrab1 3098 . . . 4 𝑦{𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4 txtop 21124 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
54adantr 479 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
6 simprl 789 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 ∈ (𝐽 ×t 𝐾))
7 eqid 2609 . . . . . . . . . . . . 13 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
87eltopss 20479 . . . . . . . . . . . 12 (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → 𝑅 (𝐽 ×t 𝐾))
95, 6, 8syl2anc 690 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 (𝐽 ×t 𝐾))
10 imasnopn.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
11 eqid 2609 . . . . . . . . . . . . 13 𝐾 = 𝐾
1210, 11txuni 21147 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
1312adantr 479 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑋 × 𝐾) = (𝐽 ×t 𝐾))
149, 13sseqtr4d 3604 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝑅 ⊆ (𝑋 × 𝐾))
15 imass1 5406 . . . . . . . . . 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 5484 . . . . . . . . . 10 (𝐴𝑋 → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1817ad2antll 760 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑋 × 𝐾) “ {𝐴}) = 𝐾)
1916, 18sseqtrd 3603 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝐾)
2019sseld 3566 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 𝐾))
2120pm4.71rd 664 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴}))))
22 vex 3175 . . . . . . . . 9 𝑦 ∈ V
23 elimasng 5397 . . . . . . . . 9 ((𝐴𝑋𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2422, 23mpan2 702 . . . . . . . 8 (𝐴𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2524ad2antll 760 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2625anbi2d 735 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑦 𝐾𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
2721, 26bitrd 266 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
28 rabid 3094 . . . . 5 (𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
2927, 28syl6bbr 276 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
301, 2, 3, 29eqrd 3585 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
31 eqid 2609 . . . 4 (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩)
3231mptpreima 5531 . . 3 ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
3330, 32syl6eqr 2661 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) = ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
3411toptopon 20490 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3534biimpi 204 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
3635ad2antlr 758 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐾 ∈ (TopOn‘ 𝐾))
3710toptopon 20490 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3837biimpi 204 . . . . . 6 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
3938ad2antrr 757 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
40 simprr 791 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → 𝐴𝑋)
4136, 39, 40cnmptc 21217 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾𝐴) ∈ (𝐾 Cn 𝐽))
4236cnmptid 21216 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾𝑦) ∈ (𝐾 Cn 𝐾))
4336, 41, 42cnmpt1t 21220 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
44 cnima 20821 . . 3 (((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
4543, 6, 44syl2anc 690 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → ((𝑦 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
4633, 45eqeltrd 2687 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  wss 3539  {csn 4124  cop 4130   cuni 4366  cmpt 4637   × cxp 5026  ccnv 5027  cima 5031  cfv 5790  (class class class)co 6527  Topctop 20459  TopOnctopon 20460   Cn ccn 20780   ×t ctx 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-topgen 15873  df-top 20463  df-bases 20464  df-topon 20465  df-cn 20783  df-cnp 20784  df-tx 21117
This theorem is referenced by: (None)
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