Theorem cnmpt2k 22293

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 2955	. . . . 5 ⊢ Ⅎ𝑥𝑌
2		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑥𝑣
3		nfmpo2 7214	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
4		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfov 7165	. . . . 5 ⊢ Ⅎ𝑥(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
6	1, 5	nfmpt 5127	. . . 4 ⊢ Ⅎ𝑥(𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
7		nfcv 2955	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
8		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
9		nfmpo1 7213	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
10		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
11	8, 9, 10	nfov 7165	. . . . . 6 ⊢ Ⅎ𝑦(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
12		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑣(𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
13		oveq1 7142	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
14	11, 12, 13	cbvmpt 5131	. . . . 5 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
15		oveq2 7143	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
16	15	mpteq2dv 5126	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
17	14, 16	syl5eq 2845	. . . 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 5131	. . 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 488	. . . . . 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 22196	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
24	21, 22, 23	syl2anc 587	. . . . . . . . . . 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 21846	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
28		toptopon2 21523	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
29	27, 28	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
30	22, 21, 25	cnmptcom 22283	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31		cnf2 21854	. . . . . . . . . . 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 1368	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
33		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
34	33	fmpo 7748	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
35	32, 34	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
36	35	r19.21bi 3173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
37	36	r19.21bi 3173	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿)
38	37	an32s 651	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
39	33	ovmpt4g 7276	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
40	19, 20, 38, 39	syl3anc 1368	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
41	40	mpteq2dva 5125	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) = (𝑦 ∈ 𝑌 ↦ 𝐴))
42	41	mpteq2dva 5125	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
43	18, 42	syl5eq 2845	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
44		eqid 2798	. . . . 5 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩))
45	44	xkoinjcn 22292	. . . 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 587	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
47	32	feqmptd 6708	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)))
48	47, 30	eqeltrrd 2891	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49		fveq2 6645	. . . 4 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩))
50		df-ov 7138	. . . 4 ⊢ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩)
51	49, 50	eqtr4di 2851	. . 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 22286	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
53	43, 52	eqeltrrd 2891	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟨cop 4531  ∪ cuni 4800   ↦ cmpt 5110   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Topctop 21498  TopOnctopon 21515   Cn ccn 21829   ×_t ctx 22165   ↑_ko cxko 22166

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cnp 21833  df-cmp 21992  df-tx 22167  df-xko 22168

This theorem is referenced by:  xkocnv  22419  xkohmeo  22420