Theorem imasnopn 23719

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) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)

Proof of Theorem imasnopn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1913	. . . 4 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋))
2		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
3		nfrab1 3464	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4		txtop 23598	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
5	4	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
6		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ (𝐽 ×t 𝐾))
7		eqid 2740	. . . . . . . . . . . . 13 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
8	7	eltopss 22934	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
9	5, 6, 8	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
10		imasnopn.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
11		eqid 2740	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
12	10, 11	txuni 23621	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
13	12	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
14	9, 13	sseqtrrd 4050	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × ∪ 𝐾))
15		imass1 6131	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝑋 × ∪ 𝐾) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
17		xpimasn 6216	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
18	17	ad2antll 728	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
19	16, 18	sseqtrd 4049	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ∪ 𝐾)
20	19	sseld 4007	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ ∪ 𝐾))
21	20	pm4.71rd 562	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
22		elimasng 6118	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
23	22	elvd 3494	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
24	23	ad2antll 728	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
25	24	anbi2d 629	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
26	21, 25	bitrd 279	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
27		rabid 3465	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
28	26, 27	bitr4di 289	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
29	1, 2, 3, 28	eqrd 4028	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
30		eqid 2740	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩)
31	30	mptpreima 6269	. . 3 ⊢ (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
32	29, 31	eqtr4di 2798	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
33	11	toptopon 22944	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
34	33	biimpi 216	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
35	34	ad2antlr 726	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
36	10	toptopon 22944	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
37	36	biimpi 216	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
38	37	ad2antrr 725	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
39		simprr 772	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
40	35, 38, 39	cnmptc 23691	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
41	35	cnmptid 23690	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
42	35, 40, 41	cnmpt1t 23694	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
43		cnima 23294	. . 3 ⊢ (((𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
44	42, 6, 43	syl2anc 583	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
45	32, 44	eqeltrd 2844	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ cuni 4931   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699   “ cima 5703  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937   Cn ccn 23253   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-cn 23256  df-cnp 23257  df-tx 23591

This theorem is referenced by: (None)