Theorem cnextfval 24198

Description: The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)

Hypotheses

Ref	Expression
cnextfval.1	⊢ 𝑋 = ∪ 𝐽
cnextfval.2	⊢ 𝐵 = ∪ 𝐾

Assertion

Ref	Expression
cnextfval	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))

Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cnextfval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextval 24197	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
2	1	adantr 485	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
3		simpr 489	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
4	3	dmeqd 5895	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5		simplrl 788	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → 𝐹:𝐴⟶𝐵)
6	5	fdmd 6716	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
7	4, 6	eqtrd 2796	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
8	7	fveq2d 6885	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ((cls‘𝐽)‘dom 𝑓) = ((cls‘𝐽)‘𝐴))
9	7	oveq2d 7426	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
10	9	oveq2d 7426	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)))
11	10, 3	fveq12d 6888	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
12	11	xpeq2d 5691	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
13	8, 12	iuneq12d 4985	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
14		uniexg 7738	. . . 4 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
15	14	ad2antlr 739	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐾 ∈ V)
16		uniexg 7738	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
17	16	ad2antrr 738	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐽 ∈ V)
18		eqid 2761	. . . . . 6 ⊢ 𝐴 = 𝐴
19		cnextfval.2	. . . . . 6 ⊢ 𝐵 = ∪ 𝐾
20	18, 19	feq23i 6699	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∪ 𝐾)
21	20	biimpi 219	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶∪ 𝐾)
22	21	ad2antrl 740	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝐴⟶∪ 𝐾)
23		cnextfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
24	23	sseq2i 3965	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽)
25	24	biimpi 219	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ ∪ 𝐽)
26	25	ad2antll 741	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐴 ⊆ ∪ 𝐽)
27		elpm2r 8841	. . 3 ⊢ (((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) ∧ (𝐹:𝐴⟶∪ 𝐾 ∧ 𝐴 ⊆ ∪ 𝐽)) → 𝐹 ∈ (∪ 𝐾 ↑pm ∪ 𝐽))
28	15, 17, 22, 26, 27	syl22anc 851	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐹 ∈ (∪ 𝐾 ↑pm ∪ 𝐽))
29		fvex 6894	. . . 4 ⊢ ((cls‘𝐽)‘𝐴) ∈ V
30		vsnex 5406	. . . . 5 ⊢ {𝑥} ∈ V
31		fvex 6894	. . . . 5 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V
32	30, 31	xpex 7751	. . . 4 ⊢ ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
33	29, 32	iunex 7964	. . 3 ⊢ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
34	33	a1i 11	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V)
35	2, 13, 28, 34	fvmptd 6997	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191   × cxp 5659  dom cdm 5661  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8824   ↾_t crest 17472  Topctop 23029  clsccl 23154  neicnei 23233   fLimf cflf 24071  CnExtccnext 24195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-pm 8826  df-cnext 24196

This theorem is referenced by:  cnextrel  24199  cnextfun  24200  cnextfvval  24201  cnextf  24202  cnextfres  24205