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Theorem cnfval 21560
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 21554 . . 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 761 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
43unieqd 4718 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
5 toponuni 21241 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
65ad2antlr 715 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑌 = 𝐾)
74, 6eqtr4d 2810 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝑌)
8 simprl 759 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
98unieqd 4718 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
10 toponuni 21241 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1110ad2antrr 714 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
129, 11eqtr4d 2810 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
137, 12oveq12d 6992 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ( 𝑘𝑚 𝑗) = (𝑌𝑚 𝑋))
148eleq2d 2844 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ((𝑓𝑦) ∈ 𝑗 ↔ (𝑓𝑦) ∈ 𝐽))
153, 14raleqbidv 3334 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗 ↔ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽))
1613, 15rabeqbidv 3401 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗} = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
17 topontop 21240 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1817adantr 473 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
19 topontop 21240 . . 3 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2019adantl 474 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
21 ovex 7006 . . . 4 (𝑌𝑚 𝑋) ∈ V
2221rabex 5087 . . 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 7116 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 387   = wceq 1508  wcel 2051  wral 3081  {crab 3085  Vcvv 3408   cuni 4708  ccnv 5402  cima 5406  cfv 6185  (class class class)co 6974  cmpo 6976  𝑚 cmap 8204  Topctop 21220  TopOnctopon 21237   Cn ccn 21551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-topon 21238  df-cn 21554
This theorem is referenced by:  iscn  21562  cnfex  40742
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