Theorem cnmpt2k 23653

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 2898	. . . . 5 ⊢ Ⅎ𝑥𝑌
2		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥𝑣
3		nfmpo2 7448	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
4		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfov 7397	. . . . 5 ⊢ Ⅎ𝑥(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
6	1, 5	nfmpt 5183	. . . 4 ⊢ Ⅎ𝑥(𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
7		nfcv 2898	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
8		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
9		nfmpo1 7447	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
10		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
11	8, 9, 10	nfov 7397	. . . . . 6 ⊢ Ⅎ𝑦(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
12		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑣(𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
13		oveq1 7374	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
14	11, 12, 13	cbvmpt 5187	. . . . 5 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
15		oveq2 7375	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
16	15	mpteq2dv 5179	. . . . 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 2783	. . . 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 5187	. . 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 769	. . . . . 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 23556	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
24	21, 22, 23	syl2anc 585	. . . . . . . . . . 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 23206	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
28		toptopon2 22883	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
29	27, 28	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
30	22, 21, 25	cnmptcom 23643	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31		cnf2 23214	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
33		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
34	33	fmpo 8021	. . . . . . . . . 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 653	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
39	33	ovmpt4g 7514	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
40	19, 20, 38, 39	syl3anc 1374	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
41	40	mpteq2dva 5178	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) = (𝑦 ∈ 𝑌 ↦ 𝐴))
42	41	mpteq2dva 5178	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
43	18, 42	eqtrid 2783	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
44		eqid 2736	. . . . 5 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩))
45	44	xkoinjcn 23652	. . . 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 585	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
47	32	feqmptd 6908	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)))
48	47, 30	eqeltrrd 2837	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49		fveq2 6840	. . . 4 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩))
50		df-ov 7370	. . . 4 ⊢ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩)
51	49, 50	eqtr4di 2789	. . 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 23646	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
53	43, 52	eqeltrrd 2837	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟨cop 4573  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Topctop 22858  TopOnctopon 22875   Cn ccn 23189   ×_t ctx 23525   ↑_ko cxko 23526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-1o 8405  df-2o 8406  df-map 8775  df-en 8894  df-dom 8895  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cn 23192  df-cnp 23193  df-cmp 23352  df-tx 23527  df-xko 23528

This theorem is referenced by:  xkocnv  23779  xkohmeo  23780