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Theorem cnfval 13697
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Cn 𝐾) = {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽})
Distinct variable groups:   𝑦,𝑓,𝐾   𝑓,𝑋,𝑦   𝑓,π‘Œ,𝑦   𝑓,𝐽,𝑦
Proof of Theorem cnfval
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cn 13691 . . 3 Cn = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ (◑𝑓 β€œ 𝑦) ∈ 𝑗})
21a1i 9 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ Cn = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ (◑𝑓 β€œ 𝑦) ∈ 𝑗}))
3 simprr 531 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘˜ = 𝐾)
43unieqd 3821 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
5 toponuni 13518 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
65ad2antlr 489 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘Œ = βˆͺ 𝐾)
74, 6eqtr4d 2213 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = π‘Œ)
8 simprl 529 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑗 = 𝐽)
98unieqd 3821 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
10 toponuni 13518 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1110ad2antrr 488 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑋 = βˆͺ 𝐽)
129, 11eqtr4d 2213 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = 𝑋)
137, 12oveq12d 5893 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) = (π‘Œ β†‘π‘š 𝑋))
148eleq2d 2247 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ ((◑𝑓 β€œ 𝑦) ∈ 𝑗 ↔ (◑𝑓 β€œ 𝑦) ∈ 𝐽))
153, 14raleqbidv 2685 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆ€π‘¦ ∈ π‘˜ (◑𝑓 β€œ 𝑦) ∈ 𝑗 ↔ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽))
1613, 15rabeqbidv 2733 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘¦ ∈ π‘˜ (◑𝑓 β€œ 𝑦) ∈ 𝑗} = {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽})
17 topontop 13517 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
1817adantr 276 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ Top)
19 topontop 13517 . . 3 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2019adantl 277 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ Top)
21 fnmap 6655 . . . 4 β†‘π‘š Fn (V Γ— V)
22 toponmax 13528 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
2322elexd 2751 . . . . 5 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ V)
2423adantl 277 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ π‘Œ ∈ V)
25 toponmax 13528 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2625elexd 2751 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ V)
2726adantr 276 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑋 ∈ V)
28 fnovex 5908 . . . 4 (( β†‘π‘š Fn (V Γ— V) ∧ π‘Œ ∈ V ∧ 𝑋 ∈ V) β†’ (π‘Œ β†‘π‘š 𝑋) ∈ V)
2921, 24, 27, 28mp3an2i 1342 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘Œ β†‘π‘š 𝑋) ∈ V)
30 rabexg 4147 . . 3 ((π‘Œ β†‘π‘š 𝑋) ∈ V β†’ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽} ∈ V)
3129, 30syl 14 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽} ∈ V)
322, 16, 18, 20, 31ovmpod 6002 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 Cn 𝐾) = {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 (◑𝑓 β€œ 𝑦) ∈ 𝐽})
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  {crab 2459  Vcvv 2738  βˆͺ cuni 3810   Γ— cxp 4625  β—‘ccnv 4626   β€œ cima 4630   Fn wfn 5212  β€˜cfv 5217  (class class class)co 5875   ∈ cmpo 5877   β†‘π‘š cmap 6648  Topctop 13500  TopOnctopon 13513   Cn ccn 13688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-top 13501  df-topon 13514  df-cn 13691
This theorem is referenced by:  cnovex  13699  iscn  13700
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