Theorem cnmptkp 23794

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 2765	. . . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
5	2	eleq1d 2850	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
6		cnmptk1.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
7	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8		cnmptk1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
9		topontop 23027	. . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ Top)
12		toptopon2 23032	. . . . . . . 8 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
13	11, 12	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿))
14		cnmptk1.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15		topontop 23027	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
16	6, 15	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Top)
17		eqid 2765	. . . . . . . . . . 11 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
18	17	xkotopon 23714	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
19	16, 10, 18	syl2anc 595	. . . . . . . . 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 23363	. . . . . . . . 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 1394	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
23	22	fvmptelcdm 7098	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
24		cnf2 23363	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
25	7, 13, 23, 24	syl3anc 1394	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
26	1	fmpt 7095	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
27	25, 26	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
28	5, 27, 4	rspcdva 3585	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
29	1, 2, 4, 28	fvmptd3 7003	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
30	29	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
31		toponuni 23028	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
32	6, 31	syl 18	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
33	3, 32	eleqtrd 2867	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
34		eqid 2765	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
35	34	xkopjcn 23770	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 ∈ ∪ 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
36	16, 10, 33, 35	syl3anc 1394	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
37		fveq1 6870	. . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑤‘𝐵) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
38	14, 20, 19, 36, 37	cnmpt11 23777	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
39	30, 38	eqeltrrd 2866	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∪ cuni 4867   ↦ cmpt 5185  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Topctop 23007  TopOnctopon 23024   Cn ccn 23338   ↑_ko cxko 23675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-1o 8441  df-2o 8442  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-fin 8935  df-fi 9359  df-rest 17463  df-topgen 17484  df-pt 17485  df-top 23008  df-topon 23025  df-bases 23060  df-cn 23341  df-cmp 23501  df-xko 23677

This theorem is referenced by: (None)