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Theorem cnmptkp 23628
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 (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptkp.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
cnmptkp.b (𝜑𝐵𝑌)
cnmptkp.c (𝑦 = 𝐵𝐴 = 𝐶)
Assertion
Ref Expression
cnmptkp (𝜑 → (𝑥𝑋𝐶) ∈ (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.
StepHypRef Expression
1 eqid 2725 . . . 4 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2 cnmptkp.c . . . 4 (𝑦 = 𝐵𝐴 = 𝐶)
3 cnmptkp.b . . . . 5 (𝜑𝐵𝑌)
43adantr 479 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
52eleq1d 2810 . . . . 5 (𝑦 = 𝐵 → (𝐴 𝐿𝐶 𝐿))
6 cnmptk1.k . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘𝑌))
76adantr 479 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8 cnmptk1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑍))
9 topontop 22859 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
1110adantr 479 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐿 ∈ Top)
12 toptopon2 22864 . . . . . . . 8 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1311, 12sylib 217 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘ 𝐿))
14 cnmptk1.j . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
15 topontop 22859 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
166, 15syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
17 eqid 2725 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1817xkotopon 23548 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
1916, 10, 18syl2anc 582 . . . . . . . . 9 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20 cnmptkp.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
21 cnf2 23197 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2214, 19, 20, 21syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2322fvmptelcdm 7122 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
24 cnf2 23197 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌 𝐿)
257, 13, 23, 24syl3anc 1368 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌 𝐿)
261fmpt 7119 . . . . . 6 (∀𝑦𝑌 𝐴 𝐿 ↔ (𝑦𝑌𝐴):𝑌 𝐿)
2725, 26sylibr 233 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 𝐿)
285, 27, 4rspcdva 3607 . . . 4 ((𝜑𝑥𝑋) → 𝐶 𝐿)
291, 2, 4, 28fvmptd3 7027 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3029mpteq2dva 5249 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
31 toponuni 22860 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
326, 31syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
333, 32eleqtrd 2827 . . . 4 (𝜑𝐵 𝐾)
34 eqid 2725 . . . . 5 𝐾 = 𝐾
3534xkopjcn 23604 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
3616, 10, 33, 35syl3anc 1368 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
37 fveq1 6895 . . 3 (𝑤 = (𝑦𝑌𝐴) → (𝑤𝐵) = ((𝑦𝑌𝐴)‘𝐵))
3814, 20, 19, 36, 37cnmpt11 23611 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
3930, 38eqeltrrd 2826 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050   cuni 4909  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856   Cn ccn 23172  ko cxko 23509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-pt 17429  df-top 22840  df-topon 22857  df-bases 22893  df-cn 23175  df-cmp 23335  df-xko 23511
This theorem is referenced by: (None)
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