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Theorem cnfval 23228
Description: The set of all continuous functions from topology 𝐽 to topology 𝐾. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnfval ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
Distinct variable groups:   𝑦,𝑓,𝐾   𝑓,𝑋,𝑦   𝑓,𝑌,𝑦   𝑓,𝐽,𝑦

Proof of Theorem cnfval
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cn 23222 . . 3 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
21a1i 11 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗}))
3 simprr 771 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
43unieqd 4926 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
5 toponuni 22907 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 725 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑌 = 𝐾)
74, 6eqtr4d 2769 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝑌)
8 simprl 769 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
98unieqd 4926 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
10 toponuni 22907 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad2antrr 724 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
129, 11eqtr4d 2769 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
137, 12oveq12d 7442 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ( 𝑘m 𝑗) = (𝑌m 𝑋))
148eleq2d 2812 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑓𝑦) ∈ 𝑗 ↔ (𝑓𝑦) ∈ 𝐽))
153, 14raleqbidv 3330 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗 ↔ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽))
1613, 15rabeqbidv 3437 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗} = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
17 topontop 22906 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1817adantr 479 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
19 topontop 22906 . . 3 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2019adantl 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
21 ovex 7457 . . . 4 (𝑌m 𝑋) ∈ V
2221rabex 5339 . . 3 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ∈ V
2322a1i 11 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ∈ V)
242, 16, 18, 20, 23ovmpod 7578 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462   cuni 4913  ccnv 5681  cima 5685  cfv 6554  (class class class)co 7424  cmpo 7426  m cmap 8855  Topctop 22886  TopOnctopon 22903   Cn ccn 23219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-topon 22904  df-cn 23222
This theorem is referenced by:  iscn  23230  cnfex  44627
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