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Theorem cnextfval 22237
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 22236 . . 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 474 . 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 479 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
43dmeqd 5559 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5 simplrl 797 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → 𝐹:𝐴𝐵)
65fdmd 6288 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
74, 6eqtrd 2862 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
87fveq2d 6438 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → ((cls‘𝐽)‘dom 𝑓) = ((cls‘𝐽)‘𝐴))
97oveq2d 6922 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
109oveq2d 6922 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) ∧ 𝑓 = 𝐹) → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)))
1110, 3fveq12d 6441 . . . 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))
1211xpeq2d 5373 . . 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)))
138, 12iuneq12d 4767 . 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)))
14 uniexg 7216 . . . 4 (𝐾 ∈ Top → 𝐾 ∈ V)
1514ad2antlr 720 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐾 ∈ V)
16 uniexg 7216 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
1716ad2antrr 719 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐽 ∈ V)
18 eqid 2826 . . . . . 6 𝐴 = 𝐴
19 cnextfval.2 . . . . . 6 𝐵 = 𝐾
2018, 19feq23i 6273 . . . . 5 (𝐹:𝐴𝐵𝐹:𝐴 𝐾)
2120biimpi 208 . . . 4 (𝐹:𝐴𝐵𝐹:𝐴 𝐾)
2221ad2antrl 721 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐹:𝐴 𝐾)
23 cnextfval.1 . . . . . 6 𝑋 = 𝐽
2423sseq2i 3856 . . . . 5 (𝐴𝑋𝐴 𝐽)
2524biimpi 208 . . . 4 (𝐴𝑋𝐴 𝐽)
2625ad2antll 722 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐴 𝐽)
27 elpm2r 8141 . . 3 ((( 𝐾 ∈ V ∧ 𝐽 ∈ V) ∧ (𝐹:𝐴 𝐾𝐴 𝐽)) → 𝐹 ∈ ( 𝐾pm 𝐽))
2815, 17, 22, 26, 27syl22anc 874 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝐹 ∈ ( 𝐾pm 𝐽))
29 fvex 6447 . . . 4 ((cls‘𝐽)‘𝐴) ∈ V
30 snex 5130 . . . . 5 {𝑥} ∈ V
31 fvex 6447 . . . . 5 ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V
3230, 31xpex 7224 . . . 4 ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
3329, 32iunex 7409 . . 3 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
3433a1i 11 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V)
352, 13, 28, 34fvmptd 6536 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 386   = wceq 1658  wcel 2166  Vcvv 3415  wss 3799  {csn 4398   cuni 4659   ciun 4741  cmpt 4953   × cxp 5341  dom cdm 5343  wf 6120  cfv 6124  (class class class)co 6906  pm cpm 8124  t crest 16435  Topctop 21069  clsccl 21194  neicnei 21273   fLimf cflf 22110  CnExtccnext 22234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-pm 8126  df-cnext 22235
This theorem is referenced by:  cnextrel  22238  cnextfun  22239  cnextfvval  22240  cnextf  22241  cnextfres  22244
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