Theorem imasnopn 15383

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 1581	. . . 4 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋))
2		nfcv 2392	. . . 4 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
3		nfrab1 2732	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
4		txtop 15344	. . . . . . . . . . . . 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 535	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ (𝐽 ×t 𝐾))
7		eqid 2238	. . . . . . . . . . . . 13 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
8	7	eltopss 15093	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ∈ (𝐽 ×t 𝐾)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
9	5, 6, 8	syl2anc 415	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
10		imasnopn.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
11		eqid 2238	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
12	10, 11	txuni 15347	. . . . . . . . . . . 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 3287	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × ∪ 𝐾))
15		imass1 5160	. . . . . . . . . 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 5234	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
18	17	ad2antll 495	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × ∪ 𝐾) “ {𝐴}) = ∪ 𝐾)
19	16, 18	sseqtrd 3286	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ∪ 𝐾)
20	19	sseld 3247	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ ∪ 𝐾))
21	20	pm4.71rd 398	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ ∪ 𝐾 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
22		elimasng 5153	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
23	22	elvd 2826	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
24	23	ad2antll 495	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
25	24	anbi2d 468	. . . . . 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 2727	. . . . 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 3266	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
30		eqid 2238	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩)
31	30	mptpreima 5279	. . 3 ⊢ (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ ∪ 𝐾 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
32	29, 31	eqtr4di 2289	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
33	11	toptopon 15102	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
34	33	biimpi 120	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
35	34	ad2antlr 493	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
36	10	toptopon 15102	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
37	36	biimpi 120	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
38	37	ad2antrr 492	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
39		simprr 537	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
40	35, 38, 39	cnmptc 15366	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
41	35	cnmptid 15365	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
42	35, 40, 41	cnmpt1t 15369	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
43		cnima 15304	. . 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 415	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (◡(𝑦 ∈ ∪ 𝐾 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) ∈ 𝐾)
45	32, 44	eqeltrd 2315	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821   ⊆ wss 3220  {csn 3708  ⟨cop 3711  ∪ cuni 3933   ↦ cmpt 4190   × cxp 4770  ^◡ccnv 4771   “ cima 4775  ‘cfv 5375  (class class class)co 6079  Topctop 15081  TopOnctopon 15094   Cn ccn 15269   ×_t ctx 15336

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-topgen 13597  df-top 15082  df-topon 15095  df-bases 15127  df-cn 15272  df-cnp 15273  df-tx 15337

This theorem is referenced by: (None)