Theorem cnmpt2k 22839

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 2907	. . . . 5 ⊢ Ⅎ𝑥𝑌
2		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑣
3		nfmpo2 7356	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
4		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfov 7305	. . . . 5 ⊢ Ⅎ𝑥(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
6	1, 5	nfmpt 5181	. . . 4 ⊢ Ⅎ𝑥(𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
7		nfcv 2907	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
8		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
9		nfmpo1 7355	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
10		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
11	8, 9, 10	nfov 7305	. . . . . 6 ⊢ Ⅎ𝑦(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
12		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑣(𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
13		oveq1 7282	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
14	11, 12, 13	cbvmpt 5185	. . . . 5 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
15		oveq2 7283	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
16	15	mpteq2dv 5176	. . . . 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 2790	. . . 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 5185	. . 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 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
20		simplr 766	. . . . . 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 22742	. . . . . . . . . . . 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 22392	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
28		toptopon2 22067	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
29	27, 28	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
30	22, 21, 25	cnmptcom 22829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31		cnf2 22400	. . . . . . . . . . 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 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
33		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
34	33	fmpo 7908	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
35	32, 34	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
36	35	r19.21bi 3134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
37	36	r19.21bi 3134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿)
38	37	an32s 649	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
39	33	ovmpt4g 7420	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
40	19, 20, 38, 39	syl3anc 1370	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
41	40	mpteq2dva 5174	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) = (𝑦 ∈ 𝑌 ↦ 𝐴))
42	41	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
43	18, 42	eqtrid 2790	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
44		eqid 2738	. . . . 5 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩))
45	44	xkoinjcn 22838	. . . 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 6837	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)))
48	47, 30	eqeltrrd 2840	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49		fveq2 6774	. . . 4 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩))
50		df-ov 7278	. . . 4 ⊢ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩)
51	49, 50	eqtr4di 2796	. . 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 22832	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
53	43, 52	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567  ∪ cuni 4839   ↦ cmpt 5157   × cxp 5587  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711   ↑_ko cxko 22712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cnp 22379  df-cmp 22538  df-tx 22713  df-xko 22714

This theorem is referenced by:  xkocnv  22965  xkohmeo  22966