Theorem imasnopn 15026

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 1576	. . . 4 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋))
2		nfcv 2374	. . . 4 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
3		nfrab1 2713	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4		txtop 14987	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
5	4	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
6		simprl 531	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ (𝐽 ×t 𝐾))
7		eqid 2231	. . . . . . . . . . . . 13 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
8	7	eltopss 14736	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
9	5, 6, 8	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
10		imasnopn.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
11		eqid 2231	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
12	10, 11	txuni 14990	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
13	12	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
14	9, 13	sseqtrrd 3266	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × ∪ 𝐾))
15		imass1 5111	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝑋 × ∪ 𝐾) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
16	14, 15	syl 14	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
17		xpimasn 5185	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
18	17	ad2antll 491	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
19	16, 18	sseqtrd 3265	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ∪ 𝐾)
20	19	sseld 3226	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ ∪ 𝐾))
21	20	pm4.71rd 394	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
22		elimasng 5104	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
23	22	elvd 2807	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
24	23	ad2antll 491	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
25	24	anbi2d 464	. . . . . 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 188	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
27		rabid 2709	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
28	26, 27	bitr4di 198	. . . 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 3245	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
30		eqid 2231	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩)
31	30	mptpreima 5230	. . 3 ⊢ (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
32	29, 31	eqtr4di 2282	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
33	11	toptopon 14745	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
34	33	biimpi 120	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
35	34	ad2antlr 489	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
36	10	toptopon 14745	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
37	36	biimpi 120	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
38	37	ad2antrr 488	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
39		simprr 533	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
40	35, 38, 39	cnmptc 15009	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
41	35	cnmptid 15008	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
42	35, 40, 41	cnmpt1t 15012	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
43		cnima 14947	. . 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 411	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
45	32, 44	eqeltrd 2308	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {crab 2514  Vcvv 2802   ⊆ wss 3200  {csn 3669  ⟨cop 3672  ∪ cuni 3893   ↦ cmpt 4150   × cxp 4723  ^◡ccnv 4724   “ cima 4728  ‘cfv 5326  (class class class)co 6018  Topctop 14724  TopOnctopon 14737   Cn ccn 14912   ×_t ctx 14979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-topgen 13345  df-top 14725  df-topon 14738  df-bases 14770  df-cn 14915  df-cnp 14916  df-tx 14980

This theorem is referenced by: (None)