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Theorem cnextfval 22085
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:𝐴𝐵𝐴𝑋)) → ((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.
StepHypRef Expression
1 cnextval 22084 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
21adantr 466 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → (𝐽CnExt𝐾) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
3 simpr 471 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
43dmeqd 5464 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5 simplrl 754 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → 𝐹:𝐴𝐵)
6 fdm 6191 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
75, 6syl 17 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
84, 7eqtrd 2804 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
98fveq2d 6336 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → ((cls‘𝐽)‘dom 𝑓) = ((cls‘𝐽)‘𝐴))
108oveq2d 6808 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
1110oveq2d 6808 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)))
1211, 3fveq12d 6338 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
1312xpeq2d 5279 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
149, 13iuneq12d 4678 . 2 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 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)))
15 uniexg 7101 . . . 4 (𝐾 ∈ Top → 𝐾 ∈ V)
1615ad2antlr 698 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐾 ∈ V)
17 uniexg 7101 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
1817ad2antrr 697 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐽 ∈ V)
19 eqid 2770 . . . . . 6 𝐴 = 𝐴
20 cnextfval.2 . . . . . 6 𝐵 = 𝐾
2119, 20feq23i 6179 . . . . 5 (𝐹:𝐴𝐵𝐹:𝐴 𝐾)
2221biimpi 206 . . . 4 (𝐹:𝐴𝐵𝐹:𝐴 𝐾)
2322ad2antrl 699 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐹:𝐴 𝐾)
24 cnextfval.1 . . . . . 6 𝑋 = 𝐽
2524sseq2i 3777 . . . . 5 (𝐴𝑋𝐴 𝐽)
2625biimpi 206 . . . 4 (𝐴𝑋𝐴 𝐽)
2726ad2antll 700 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐴 𝐽)
28 elpm2r 8026 . . 3 ((( 𝐾 ∈ V ∧ 𝐽 ∈ V) ∧ (𝐹:𝐴 𝐾𝐴 𝐽)) → 𝐹 ∈ ( 𝐾pm 𝐽))
2916, 18, 23, 27, 28syl22anc 1476 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐹 ∈ ( 𝐾pm 𝐽))
30 fvex 6342 . . . 4 ((cls‘𝐽)‘𝐴) ∈ V
31 snex 5036 . . . . 5 {𝑥} ∈ V
32 fvex 6342 . . . . 5 ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V
3331, 32xpex 7108 . . . 4 ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
3430, 33iunex 7293 . . 3 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
3534a1i 11 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V)
362, 14, 29, 35fvmptd 6430 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  wss 3721  {csn 4314   cuni 4572   ciun 4652  cmpt 4861   × cxp 5247  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6792  pm cpm 8009  t crest 16288  Topctop 20917  clsccl 21042  neicnei 21121   fLimf cflf 21958  CnExtccnext 22082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-pm 8011  df-cnext 22083
This theorem is referenced by:  cnextrel  22086  cnextfun  22087  cnextfvval  22088  cnextf  22089  cnextfres  22092
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