Theorem cnmptkp 23727

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 2761	. . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
5	2	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
6		cnmptk1.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
7	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8		cnmptk1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
9		topontop 22960	. . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
11	10	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ Top)
12		toptopon2 22965	. . . . . . . 8 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
13	11, 12	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿))
14		cnmptk1.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15		topontop 22960	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
16	6, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Top)
17		eqid 2761	. . . . . . . . . . 11 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
18	17	xkotopon 23647	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
19	16, 10, 18	syl2anc 593	. . . . . . . . 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 23296	. . . . . . . . 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 1389	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
23	22	fvmptelcdm 7088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
24		cnf2 23296	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
25	7, 13, 23, 24	syl3anc 1389	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
26	1	fmpt 7085	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
27	25, 26	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
28	5, 27, 4	rspcdva 3581	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
29	1, 2, 4, 28	fvmptd3 6993	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
30	29	mpteq2dva 5190	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
31		toponuni 22961	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
32	6, 31	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
33	3, 32	eleqtrd 2863	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
34		eqid 2761	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
35	34	xkopjcn 23703	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 ∈ ∪ 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
36	16, 10, 33, 35	syl3anc 1389	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ↑ko 𝐾) Cn 𝐿))
37		fveq1 6860	. . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑤‘𝐵) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
38	14, 20, 19, 36, 37	cnmpt11 23710	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
39	30, 38	eqeltrrd 2862	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∪ cuni 4862   ↦ cmpt 5178  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  Topctop 22940  TopOnctopon 22957   Cn ccn 23271   ↑_ko cxko 23608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-1o 8430  df-2o 8431  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-fin 8924  df-fi 9350  df-rest 17441  df-topgen 17462  df-pt 17463  df-top 22941  df-topon 22958  df-bases 22993  df-cn 23274  df-cmp 23434  df-xko 23610

This theorem is referenced by: (None)