Theorem cnmptkp 21483

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		cnmptkp.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑌)
3		cnmptk1.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
5		cnmptk1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
6		topontop 20718	. . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
8	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ Top)
9		eqid 2622	. . . . . . . . 9 ⊢ ∪ 𝐿 = ∪ 𝐿
10	9	toptopon 20722	. . . . . . . 8 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
11	8, 10	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿))
12		cnmptk1.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
13		topontop 20718	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
14	3, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ Top)
15		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
16	15	xkotopon 21403	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
17	14, 7, 16	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
18		cnmptkp.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
19		cnf2 21053	. . . . . . . . . 10 ⊢ ((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))
20	12, 17, 18, 19	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
21		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
22	21	fmpt 6381	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
23	20, 22	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
24	23	r19.21bi 2932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
25		cnf2 21053	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
26	4, 11, 24, 25	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
27		eqid 2622	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
28	27	fmpt 6381	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶∪ 𝐿)
29	26, 28	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
30		cnmptkp.c	. . . . . . 7 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
31	30	eleq1d 2686	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
32	31	rspcv 3305	. . . . 5 ⊢ (𝐵 ∈ 𝑌 → (∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 → 𝐶 ∈ ∪ 𝐿))
33	2, 29, 32	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
34	30, 27	fvmptg 6280	. . . 4 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝐿) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
35	2, 33, 34	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵) = 𝐶)
36	35	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
37		toponuni 20719	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
38	3, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
39	1, 38	eleqtrd 2703	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
40		eqid 2622	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
41	40	xkopjcn 21459	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 ∈ ∪ 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ^ko 𝐾) Cn 𝐿))
42	14, 7, 39, 41	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤‘𝐵)) ∈ ((𝐿 ^ko 𝐾) Cn 𝐿))
43		fveq1 6190	. . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → (𝑤‘𝐵) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵))
44	12, 18, 17, 42, 43	cnmpt11 21466	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
45	36, 44	eqeltrrd 2702	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∪ cuni 4436   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^_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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-xko 21366

This theorem is referenced by: (None)