Theorem cnmpt2k 23575

Description: The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypotheses

Ref	Expression
cnmpt2k.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt2k.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt2k.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Assertion

Ref	Expression
cnmpt2k	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2k

Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑌
2		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑣
3		nfmpo2 7470	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
4		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfov 7417	. . . . 5 ⊢ Ⅎ𝑥(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
6	1, 5	nfmpt 5205	. . . 4 ⊢ Ⅎ𝑥(𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
7		nfcv 2891	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
8		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
9		nfmpo1 7469	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
10		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
11	8, 9, 10	nfov 7417	. . . . . 6 ⊢ Ⅎ𝑦(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
12		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑣(𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
13		oveq1 7394	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
14	11, 12, 13	cbvmpt 5209	. . . . 5 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
15		oveq2 7395	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
16	15	mpteq2dv 5201	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
17	14, 16	eqtrid 2776	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
18	6, 7, 17	cbvmpt 5209	. . 3 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
19		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
20		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
21		cnmpt2k.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
22		cnmpt2k.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
23		txtopon 23478	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
24	21, 22, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25		cnmpt2k.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26		cntop2 23128	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
28		toptopon2 22805	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
29	27, 28	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
30	22, 21, 25	cnmptcom 23565	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31		cnf2 23136	. . . . . . . . . . 11 ⊢ (((𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
32	24, 29, 30, 31	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
33		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
34	33	fmpo 8047	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
35	32, 34	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
36	35	r19.21bi 3229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
37	36	r19.21bi 3229	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿)
38	37	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
39	33	ovmpt4g 7536	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
40	19, 20, 38, 39	syl3anc 1373	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
41	40	mpteq2dva 5200	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) = (𝑦 ∈ 𝑌 ↦ 𝐴))
42	41	mpteq2dva 5200	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
43	18, 42	eqtrid 2776	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
44		eqid 2729	. . . . 5 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩))
45	44	xkoinjcn 23574	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
46	22, 21, 45	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
47	32	feqmptd 6929	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)))
48	47, 30	eqeltrrd 2829	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49		fveq2 6858	. . . 4 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩))
50		df-ov 7390	. . . 4 ⊢ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩)
51	49, 50	eqtr4di 2782	. . 3 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
52	22, 21, 24, 46, 48, 51	cnmptk1 23568	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
53	43, 52	eqeltrrd 2829	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595  ∪ cuni 4871   ↦ cmpt 5188   × cxp 5636  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Topctop 22780  TopOnctopon 22797   Cn ccn 23111   ×_t ctx 23447   ↑_ko cxko 23448

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-2o 8435  df-map 8801  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cn 23114  df-cnp 23115  df-cmp 23274  df-tx 23449  df-xko 23450

This theorem is referenced by:  xkocnv  23701  xkohmeo  23702