Theorem cnmpt2k 21491

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 2764	. . . . 5 ⊢ Ⅎ𝑥𝑌
2		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝑣
3		nfmpt22 6723	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
4		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfov 6676	. . . . 5 ⊢ Ⅎ𝑥(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
6	1, 5	nfmpt 4746	. . . 4 ⊢ Ⅎ𝑥(𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
7		nfcv 2764	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
8		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
9		nfmpt21 6722	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
10		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
11	8, 9, 10	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑦(𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
12		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑣(𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
13		oveq1 6657	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
14	11, 12, 13	cbvmpt 4749	. . . . 5 ⊢ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
15		oveq2 6658	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
16	15	mpteq2dv 4745	. . . . 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 2668	. . . 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 4749	. . 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 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
20		simplr 792	. . . . . 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 21394	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
24	21, 22, 23	syl2anc 693	. . . . . . . . . . 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 21045	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
28		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ 𝐿 = ∪ 𝐿
29	28	toptopon 20722	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
30	27, 29	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
31	22, 21, 25	cnmptcom 21481	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
32		cnf2 21053	. . . . . . . . . . 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)
33	24, 30, 31, 32	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
34		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
35	34	fmpt2 7237	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
36	33, 35	sylibr 224	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
37	36	r19.21bi 2932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
38	37	r19.21bi 2932	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿)
39	38	an32s 846	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
40	34	ovmpt4g 6783	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
41	19, 20, 39, 40	syl3anc 1326	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
42	41	mpteq2dva 4744	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) = (𝑦 ∈ 𝑌 ↦ 𝐴))
43	42	mpteq2dva 4744	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
44	18, 43	syl5eq 2668	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)))
45		eqid 2622	. . . . 5 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩))
46	45	xkoinjcn 21490	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
47	22, 21, 46	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
48	33	feqmptd 6249	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)))
49	48, 31	eqeltrrd 2702	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
50		fveq2 6191	. . . 4 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩))
51		df-ov 6653	. . . 4 ⊢ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤) = ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘⟨𝑣, 𝑤⟩)
52	50, 51	syl6eqr 2674	. . 3 ⊢ (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
53	22, 21, 24, 47, 49, 52	cnmptk1 21484	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋 ↦ (𝑣 ∈ 𝑌 ↦ (𝑣(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
54	44, 53	eqeltrrd 2702	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363   ^_ko cxko 21364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-xko 21366

This theorem is referenced by:  xkocnv  21617  xkohmeo  21618