Theorem imasncld 23585

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)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))

Proof of Theorem imasncld

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . . 4 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋))
2		nfcv 2892	. . . 4 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
3		nfrab1 3429	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)))
5		eqid 2730	. . . . . . . . . . . . 13 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
6	5	cldss 22923	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
7	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
8		imasnopn.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
9		eqid 2730	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
10	8, 9	txuni 23486	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
11	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × ∪ 𝐾) = ∪ (𝐽 ×t 𝐾))
12	7, 11	sseqtrrd 3987	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × ∪ 𝐾))
13		imass1 6075	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝑋 × ∪ 𝐾) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × ∪ 𝐾) “ {𝐴}))
15		xpimasn 6161	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
16	15	ad2antll 729	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
17	14, 16	sseqtrd 3986	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ∪ 𝐾)
18	17	sseld 3948	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ ∪ 𝐾))
19	18	pm4.71rd 562	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
20		elimasng 6063	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
21	20	elvd 3456	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
22	21	ad2antll 729	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
23	22	anbi2d 630	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
24	19, 23	bitrd 279	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
25		rabid 3430	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ ∪ 𝐾 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
26	24, 25	bitr4di 289	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
27	1, 2, 3, 26	eqrd 3969	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
28		eqid 2730	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩)
29	28	mptpreima 6214	. . 3 ⊢ (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
30	27, 29	eqtr4di 2783	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
31	9	toptopon 22811	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
32	31	biimpi 216	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
33	32	ad2antlr 727	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
34	8	toptopon 22811	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
35	34	biimpi 216	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
36	35	ad2antrr 726	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
37		simprr 772	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
38	33, 36, 37	cnmptc 23556	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
39	33	cnmptid 23555	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
40	33, 38, 39	cnmpt1t 23559	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
41		cnclima 23162	. . 3 ⊢ (((𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾))) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
42	40, 4, 41	syl2anc 584	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ (Clsd‘𝐾))
43	30, 42	eqeltrd 2829	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ cuni 4874   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640   “ cima 5644  ‘cfv 6514  (class class class)co 7390  Topctop 22787  TopOnctopon 22804  Clsdccld 22910   Cn ccn 23118   ×_t ctx 23454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-cn 23121  df-cnp 23122  df-tx 23456

This theorem is referenced by: (None)