Theorem cnfval 13073

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 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 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.

Step	Hyp	Ref	Expression
1		df-cn 13067	. . 3 ⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗})
2	1	a1i 9	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}))
3		simprr 528	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑘 = 𝐾)
4	3	unieqd 3808	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑘 = ∪ 𝐾)
5		toponuni 12892	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
6	5	ad2antlr 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑌 = ∪ 𝐾)
7	4, 6	eqtr4d 2207	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑘 = 𝑌)
8		simprl 527	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑗 = 𝐽)
9	8	unieqd 3808	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = ∪ 𝐽)
10		toponuni 12892	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
11	10	ad2antrr 486	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → 𝑋 = ∪ 𝐽)
12	9, 11	eqtr4d 2207	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ∪ 𝑗 = 𝑋)
13	7, 12	oveq12d 5875	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (∪ 𝑘 ↑𝑚 ∪ 𝑗) = (𝑌 ↑𝑚 𝑋))
14	8	eleq2d 2241	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → ((◡𝑓 “ 𝑦) ∈ 𝑗 ↔ (◡𝑓 “ 𝑦) ∈ 𝐽))
15	3, 14	raleqbidv 2678	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → (∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗 ↔ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽))
16	13, 15	rabeqbidv 2726	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑘 = 𝐾)) → {𝑓 ∈ (∪ 𝑘 ↑𝑚 ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗} = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
17		topontop 12891	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
18	17	adantr 274	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
19		topontop 12891	. . 3 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
20	19	adantl 275	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
21		fnmap 6637	. . . 4 ⊢ ↑𝑚 Fn (V × V)
22		toponmax 12902	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
23	22	elexd 2744	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
24	23	adantl 275	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑌 ∈ V)
25		toponmax 12902	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
26	25	elexd 2744	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
27	26	adantr 274	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑋 ∈ V)
28		fnovex 5890	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌 ↑𝑚 𝑋) ∈ V)
29	21, 24, 27, 28	mp3an2i 1338	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑌 ↑𝑚 𝑋) ∈ V)
30		rabexg 4133	. . 3 ⊢ ((𝑌 ↑𝑚 𝑋) ∈ V → {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
31	29, 30	syl 14	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V)
32	2, 16, 18, 20, 31	ovmpod 5984	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1349   ∈ wcel 2142  ∀wral 2449  {crab 2453  Vcvv 2731  ∪ cuni 3797   × cxp 4610  ^◡ccnv 4611   “ cima 4615   Fn wfn 5195  ‘cfv 5200  (class class class)co 5857   ∈ cmpo 5859   ↑_𝑚 cmap 6630  Topctop 12874  TopOnctopon 12887   Cn ccn 13064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-fv 5208  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-map 6632  df-top 12875  df-topon 12888  df-cn 13067

This theorem is referenced by:  cnovex  13075  iscn  13076