Theorem cnmptkp 23670

Description: The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptk1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmptkp.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptkp.b	⊢ (𝜑 → 𝐵 ∈ 𝑌)
cnmptkp.c	⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)

Assertion

Ref	Expression
cnmptkp	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑍,𝑦   𝑥,𝐵   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑦,𝐵   𝑦,𝐶

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem cnmptkp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . 4 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
2		cnmptkp.c	. . . 4 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
3		cnmptkp.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
5	2	eleq1d 2825	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
6		cnmptk1.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
7	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8		cnmptk1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
9		topontop 22903	. . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
11	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ Top)
12		toptopon2 22908	. . . . . . . 8 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
13	11, 12	sylib 219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿))
14		cnmptk1.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15		topontop 22903	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
16	6, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Top)
17		eqid 2740	. . . . . . . . . . 11 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
18	17	xkotopon 23590	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
19	16, 10, 18	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20		cnmptkp.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
21		cnf2 23239	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
22	14, 19, 20, 21	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
23	22	fvmptelcdm 7061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
24		cnf2 23239	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
25	7, 13, 23, 24	syl3anc 1379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
26	1	fmpt 7058	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
27	25, 26	sylibr 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
28	5, 27, 4	rspcdva 3568	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
29	1, 2, 4, 28	fvmptd3 6966	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
30	29	mpteq2dva 5172	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
31		toponuni 22904	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
32	6, 31	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
33	3, 32	eleqtrd 2842	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
34		eqid 2740	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
35	34	xkopjcn 23646	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 ∈ ∪ 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
36	16, 10, 33, 35	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
37		fveq1 6833	. . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑤‘𝐵) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
38	14, 20, 19, 36, 37	cnmpt11 23653	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
39	30, 38	eqeltrrd 2841	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∪ cuni 4845   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Topctop 22883  TopOnctopon 22900   Cn ccn 23214   ↑_ko cxko 23551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-1o 8402  df-2o 8403  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-fin 8894  df-fi 9321  df-rest 17383  df-topgen 17404  df-pt 17405  df-top 22884  df-topon 22901  df-bases 22936  df-cn 23217  df-cmp 23377  df-xko 23553

This theorem is referenced by: (None)